Hello everyone, hello patrons and happy new year to you. Happy 2023 and thank you very much for your support over the last 12 months. Thank you very much. A little bit of an announcement just from Jeff and I on what's happening going forward. So you probably have noticed on your patron feed that your billing has been temporarily paused while Jeff and I rethought our plans for this year. We're largely going to go back to a predominantly free version of the podcast focused around YouTube primarily but also via Instagram and Twitter. Yeah so you can reach us on our old handles at the agile podcast on Instagram but also on LinkedIn.  That's how we missed just coming on. YouTube. Yeah it's not LinkedIn, screw LinkedIn. YouTube. That's the place to be. Yeah so we really found you support and we hope that you'll want to carry on supporting us. What we're going to do is we're going to simplify it, we're going to scale it back, we're going to have just one tier basically and not really making any promises to you about what's going to be on there. Any exclusives, any behind the scenes, any extra episodes, any bonuses of course it will come to you and as we are experimenting with lots of different ways of running structuring this podcast which in our first couple of episodes you'll see what we mean then we will be looking for feedback, we'll be looking to learn what our audience like and don't like and you will be our primary audience to find out that and inspect in depth. Yeah very good so watch this space and thank you again once again for all your support last year and let's look forward to we hope you like what's coming this year. All the best. Cheers.


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Football still isn't coming home for England's socially conscious team have done the country proud and potentially changed it for the better. Sadly the same cannot be said for that for the minority of fans who after the game flooded the social media pages of free black players with racist abuse on tonight's tisgy so we'll be discussing the politics of England's remarkable Euro 2020 campaign what it says about the country and its prospects for the future. To do so I will be joined in one moment by Aaron bus Stani and of course as ever we want to hear from you as well do you think England's anti-racist football team have changed England for the better or do the racist comments directed at Saka Sancho and Rashford show any attempt to create an anti-racist Englishness is a losing game let us know your response. By tweeting on the hashtag tisgy sour if you're new to the show hit subscribe will also be talking in the second half of the program about the latest on the loosening of coronavirus restrictions a big announcement today in parliament from such a Javed and then also a press conference from Boris Johnson first story though. England's inspirational run in Euro 2020 ended in heartbreak last night as Italy won the final at Wembley on penalties we can't show you those final moments of the game that's for copyright reasons we can't yet afford football rights but here instead is the fan reaction to the moment 22 year old Jan Luigi donnaruma saved a penalty from 19 year old. Saka now those looks of shock and disappointment were mirrored across England I'm sure many of you watching this look like that last night yet despite the sorrow I think the overwhelming mood across the country was incredibly supportive of England's play as much as we're like devastated we're proud of Saka and everything you do and everything the team achieve is I mean like it's you know it's a moment to remember we got to the final. It's very sad but we did our best and you know what would your message be to go to the players. I mean we love you. We love you. And you did us proud. Those like that years and how we came into my school you know I feel I feel attached to him in that way and you know I think it was just really unlucky they went to penalties. It's all about love. Yeah I think you know we thought we had it. As I say I think that is reflective of most people in Britain incredibly supportive of a team who got that far and did it in a really honourable fashion. However as I assume you know by this point such supportive sentiments couldn't protect. Jayden Sancho and Marcus Rashford the free players who missed penalties from online racist abuse. Now all three players received streams of racist comments and monkey emojis really horrible stuff leading to Saka suspending his Instagram account. I think that's live again now but obviously it was incredibly upsetting for the man soon after that defeat. Also we've seen a mural of Marcus Rashford which was defaced in Manchester that's now been covered and with more positive messages. There's clearly you know a big fight back against these racists as well. Now these examples I've just given you over racism everyone in mainstream politics across the political spectrum and across sport has been condemning it. I want to show you how Gareth Southgate responded to the abuse suffered by his black players. We can only set the example that we believe we should and represent the country in the way that we feel you should when you're representing England and everybody has to remember when they support the team that they also represent England and should represent what we stand for. So I think the players have done that brilliantly and we can only continue to try to affect the things that we can. But we have I think had a positive effect on lots of areas of society but we can't affect everything. Other people have responsibilities in those areas and you know we've all got to work collectively to constantly improve those things. There's a characteristically fought for intervention from Gareth Southgate. He's saying the players have set an example that's had a positive effect but there's a limit to what they can do. I think we should all celebrate players taking the knee. He's basically saying you can't expect that to end racism in this country. You're going to have to have different institutions intervening to stop black players after an important match suffering abuse. When they've put themselves forward and taken a huge risk by taking a penalty in the final for England. Interesting there. I think he said we can't do this on our own. I think we have Aaron ready now. Aaron what did you make of that intervention from Gareth Southgate? He's essentially saying we've done our bit. We've done our bit. We have put forward an anti-racist vision of England that has had some effect but we can't stop this minority of players abusing our black players. That's where other institutions have to come in potentially. That's the government potentially. That's social media companies. What do you make of this? It's a hard one. I think on the one hand if you're playing sort of a devil's advocate here you would say that footballers in the past have been villainised, demonised for doing the same thing. One example is David Beckham after World Cup 98. He was sent off for a petulant foul basically. One might argue he cost England the game and in the subsequent season, the first few games of the season there were effigies of the guy. You could say the same about players who've missed penalties and so on. They've all been demonised. What's changed is obviously social media. The costs of being able to get to these people and say outrageous things to them has clearly fallen. I think if Beckham had done what he'd done in 1998 I think he would have come in for a vociferous abuse. Clearly with these guys there's clearly a massive racial element to it. I think it is important to say that Michael if these were white footballers they also would have been abused. I think we shouldn't be under any illusions about that. Where I think there is something really important to say politically is how many institutions Michael can we say in our public life where we have people who are particularly black players, particularly black players. I don't think a Jewish person has ever played for England. George Cohen who was in the 1966 World Cup I think he had a great grandfather who was Jewish. I don't think we've had any British Asians played for the national side, but particularly black players who have a major role in professional football in this country throughout the football pyramid and also obviously in the national team. This is one of the very few institutions where they're this prominent where they play this larger role and I think it's inarguable. Actually people feel more disinhibited than anywhere else in terms of doling out abuse. So on the one hand they would be getting all sorts of vile comments whatever the colour of their skin because people sadly treat people with a public profile, footballers like that when these things happen social media has changed the rules of the game. At the same time it's clearly happening within a broader context and Michael I think what's really important is when you go back to say two weeks a month before this tournament started people were saying maybe you and I weren't saying it, but people who were perfectly progressive were saying oh god maybe Southgate, maybe the England team will sort of step back and won't take the knee and they don't want to be seen as overly political and actually they defied that and chose to do it anyway and I think that's a really really important choice they made and in that sense they've been completely vindicated. You absolutely have to stand up to these things. Yes it shouldn't just be for a few games it shouldn't be oh are we always going to do this? Yes why not? Why shouldn't we always see this? You know the PFA ultimately it's the Professional Footballers Association it represents those players many of those players are players of colour and they clearly feel there's major problems because there are in the national game with regards to racism and I think it's brilliant that you know we're having this conversation in a way straight away obviously it's terrible that it's had to happen at all but you know we should be under no illusions you know racism in football is a huge huge problem a huge problem and yes you know okay there's an element of this which is just social media many of these people won't be in the UK many of them will have multiple accounts etc etc but my god we all know this stuff happens in stands we've seen this happen and people getting pissed in pubs and bars when watching football you know there was Julian Rubenstein who said he came in for abuse because he had Rubenstein a Jewish name on the back of his England shirt we know this happens and I think it's really really brilliant frankly that it's not just being given carte blanche or being called out but actually it's front and centre now of the conversation around the national game fantastic that's really good but you know Michael Bacchiosakis only 19 years old and it shouldn't be on the shoulders of a 19 year old man to have to be at the crucible of this conversation which we should be having as a nation but you know that's really terrifying for him and I think this may sound strange again for our audience but Gareth Southgate ultimately chose a 19 year old kid who'd never taken a penalty in professional football to take the most important penalty in the history of the English game which is what it was we didn't win on penalties in 1966 and I think that was irresponsible because I think this was always going to happen you know Gary Nebler said this earlier on today we'll probably show that clip that the second he saw those three guys miss their penalties important to say actually I thought Rashford was actually an excellent penalty he just got unlucky here at the post he knew what the conversation was going to be today and I think for Sackler in particular you're 19 it's an appalling thing to have to go through at any age but I think particularly for him he's only had what one and a half seasons in the professional game as a kid I want to go straight to some comments because we've got some interesting ones Ishtac with a tenor says big up Sackler and the lads the team has achieved more in their short lives and these trolls and racists achieved in a hundred lifetimes let them wallow in their own filth I fully endorse that statement we've got a question for Aaron Kryptonian tweets on the hashtag Tiskey Sour Aaron you tweeted that the England team are defiantly anti-racist because their generation are increasingly and actively anti-racist because slowly step by step the anti-racist struggle is being won from the bottom up do you still think so? Yeah 100% 100% I don't think there's any doubts about it I think the fact you've got a manager who is willing to line up and defend these players I think the fact for me what summed it up Mike was when you've got Sam Alladies on GB News I don't know Sam Alladies politics maybe he's really radical I don't think he is but given his age I thought the way he responded to the question about Black Lives Matter was really subtle, nuanced, attuned and I think that really demonstrates as a profound shift in professional football and it would be hard not to because this is a game where huge, disproportionate number of professional players sadly not coaches, professional players of Afro-Caribbean descent heritage so the culture within the game I think is actually far more attuned to these things and a lot of commentators and pundits seem to think because it's working class people from Sam Alladies all the way through to Gary Neville or Jamie Carragher the sort of backgrounds they come from obviously now very wealthy people but I think the assumption is from liberal intelligentsia that they're not attuned to these conversations well actually when you work in an industry where you might be in a team of the starting 11, 5 or 6 of your teammates are people of colour, maybe more massive numbers of young fans are people of colour then I think it would be very hard not to be attuned to these issues so I do think it's definitely definitely improving and you can almost see it kind of cascading down, I was talking about this earlier too a colleague asked Michael Gary, I had a video and for me there's already a huge distinction between millennials and Gen Z in terms of their social morals on this stuff and that's just a cohort of 10 years difference if you think about the young players, the golden generation for England, Lampard, Gerard Rio Ferdinand, John Terry Michael Owen Rooney, all different ages that generation Ashley Cole that generation of players that came through you know, I think John Terry for instance, what he did what he did to Ashley Cole it was wasn't it we did to Rio Ferdinand, my apologies and Ashley Cole got him off the hook what he did to Anton Ferdinand rather racist comments he made I think that to me signified a kind of dying culture and football which they'd inherited from generations prior to them and I do think there's a clear discernible difference in values between the current England team generally 27 or younger generally Gen Z and even people in their late 30s, early 40s right, people like Beckham people like Neville and Neville's got really good politics or Jamie Carragher but the way they conduct themselves when they were players hugely different to this and so I think you know you'd have to you'd have to be blind almost to not see a huge cultural shift in football and we can have a conversation about that for me I think it's reflective of broader social changes yes let's have a look at I think some more evidence whilst there are still obviously I mean as we've seen huge problems there has been change and that is down to the struggles of people fighting against racism in football first I want to show you polling which is from opinion and it's on the issue of whether or not people support or repose England players taking the knee now at the start of the tournament they're already a majority who supported England players taking the knee despite what right wing pundits might say that ordinary working class people hate this now the majority of the country have been in favour of this since the start of the tournament and positively that has gone up throughout the contest so at the start of the tournament there was 50% of people who supported taking the knee that's gone up to 56% and of people who oppose taking the knee that's gone down from 37% to 32% so you're seeing an increasing consensus that players expressing their support for anti-racist struggles at the start of matches is the right thing to do another thing I thought was interesting this morning was the front pages of the right wing newspapers the right wing newspapers who were very used to very comfortable promoting racist dog whistles essentially they chose not to this morning so I want to show you the Daily Express they went with it hurts but we're so proud of you the sun went with pride of lions and the Daily Mail went with it all ends in tears but they talked about the team as lion hearts to the last what this is showing to me is not that Britain's right wing tabloids have suddenly converted to anti-racism but rather that these are newspapers who if they don't have any principles one thing they do know is who their readers are and we know they are not very willing to write headlines which are trying to encourage racism against black people if they think that's not going to damage their business prospects but it seems to me clear in this situation that the sun Daily Mail express they could have lent in to this idea that there were black players at miss penalties and they should have been focused on football instead of politics obviously we're going to debunk why all of this is completely ridiculous in a moment but they could have gone with that and they've decided no actually that wouldn't wash with our readers and I think that does show some progress now of course none of this is to minimize the problem because even if the racism we've seen online and in the streets I mean this is less documented even if that's from a real minority of people that doesn't mean it doesn't have a huge effect on black people in this country and black people in football I want to go to a couple of tweets which I think really made this stand out to me really hammered home this point now the first is from Marvin Sordell he's a former footballer now producer and speaker so he said on twitter the saddest thing is that as soon as free black players missed penalties we all knew exactly what was coming you can see there how ingrained this is how used to racist abuse black people are whenever a black footballer makes a mistake people always say when a black footballer succeeds that's a victory for England footballer makes a mistake that's the fault of them individually and this is something people are so used to seeing that people knew instantly when those players missed the penalty what was coming the second one is from Jason Okendaya who we've had on the show before he's a writer he shared a screenshot from his family group chat with his mum saying evening all if any of you are out please get home early I am begging you all it is going to be rowdy and violent tonight when black players missed penalties were by black players now that's a really heart breaking thing to share the fact that we still live in a nation where people are worried about their kids when black players missed penalties is shocking however much progress we've had while that is still the case there is clearly a significant problem it's also a significant problem when this racist behaviour by minorities is being boosted by people in power which is the topic of the next section before we go on to that if you're enjoying the stream do hit that like button the minority of fans who reacted to England's loss with racism have been condemned by people across the political spectrum that includes the prime minister who said this England team deserved to be lauded as heroes not racially abused on social media those responsible for this appalling abuse should be ashamed of himself now those condemnations of despicable abuse are of course welcome it would be terrible if we had a prime minister who said oh no good people on both sides whatever but do such condemnations ring hollow when they come from politicians who have pursued racist policies and made racist statements Gary Neville thinks so in the players a few weeks ago about five days on the trot told us that they were taking the need to promote equality and it was against racism the prime minister said that it was okay for the population of this country to boo those players who were trying to promote equality and defend against racism it starts at the very top and so for me I wasn't surprised in the slightest that I woke up this morning to those headlines I expected at the minute that the three players that missed missed you know the fact of the matter is there is an issue obviously in football there is an issue in society where we feel it's acceptable basically to criticise players for sporting actions because of the colour of their skin but I have to say accepting and validating that players who take the need I would say we're promoting equality inclusion and defending against racism is coming from the very top and you know full well that if your parents do something your children will follow and I'm not saying that each individual person who was obviously directed that abuse towards the players shouldn't take accountability the absolute should social and social media companies you know we know full well now it's a well told story that the social media companies have to come down harder on racist abuse I think we have to start to isolate these individuals who are attacking the players in a racist manner and isolate them by writing to their employers so that ultimately there is total accountability and there is suffering and consequence but also that to be fair there is consequence within the game through the FA the Premier League and other organisations UEFA and FIFA and we start to see you know sentences and punishments for these incidents that ultimately fit the crime because people are being abused it's absolutely ridiculous in 2021 we're still talking about this but we are so Gary Neville they're saying this comes from the top he mentioned in other interviews as well Boris Johnson's comments about Muslim women looking like letter boxes he's saying we have a racist culture which comes from the top essentially which is very different from what's being put forward by for example Boris Johnson who's saying this is just a minority of people who need to crawl back under a rock Aaron I want to get your thoughts on this do you think Gary Neville's right to say that this comes from the top yeah I think 100% I don't think there's any doubt about that I think it would be a more interesting conversation if you didn't have somebody like Boris Johnson as the Prime Minister and if you made that same point about how Britain because of course then the conversation becomes incredibly abstract doesn't it but when you've got a figurehead like a Trump or a Boris Johnson Johnson is not as bad as Trump but you know he's still pretty bad then I think it personalises things and it makes it much easier to sort of say that and say look these aren't a few bad apples these aren't isolated incidents actually they reflect a far deeper structural structural problem and I think you know in a way and this might seem quite harmless Michael but I think you can kind of look at the problem through the prism of how Harry Kane was on most front pages of most newspapers the day after England played and of course England didn't lose any of their games in normal time so you know you always had a good front page the next day had a great tournament but for me I don't think Raheem Sterling or Harry Kane played particularly well last night because they were up against two of the best centre backs in the world but for me actually Raheem Sterling had a much better tournament than Harry Kane I thought he was probably Harry Kane had a very good tournament I thought Raheem Sterling was really exciting I thought he was really working hard he's not had a great season man city I thought he was brilliant in this tournament I think if England won last night I think he would have been in the team of the tournament him and Kane both up front and yet you know how many times do we see Raheem Sterling on the front page or the back page of a newspaper not often generally speaking it was Harry Kane he was seen as the talisman of this England team and let's be honest that's because his name is Harry and because he's white Raheem Sterling is a very telegenic guy he's very affable he has a really good amazing story and yet he wasn't seen as the talisman and the simple reason for that Michael is the fact that he's black and Harry Kane is white in my opinion and so I think yes that may just seem harmless it may seem that I'll come on Aaron that's ridiculous some people might think that but I think that does show you the fact that clearly we are talking about something quite structural here and quite profound and there are lots of people Michael in the mainstream media who for instance when you had Burnley fans during the Premier League season there was a fly over the plane and I think it was like all lives matter over a game protesting black lives matter there are a lot of people in the media Michael who think yeah that's right they're fine to do that or they're entitled to do that and I don't think that's acceptable and I think that's where you say there's at the very least a culture of permissibility so on the one hand I think it's true you've got somebody like Boris Johnson who says outrageous things and I think that shows a real problem but I think for me the broader cultural problem within the media and the political class per se is permissibility how many people on the right in this country would stand up and say no that isn't acceptable when you've got a pitch and 50% of the players on the pitch are black and they're involved in an anti-racist movement which started because a black guy was killed by a police officer and they all know the reality of living with racism day in day out and some fucking idiot excuse my French pays for a plane to go over all lives matter basically saying piss off I haven't got the time of day for what you've had to live through clearly that has to be universally condemned and it wasn't right it wasn't and so I think for me the big problem is the permissibility and this is where we end up this is where we end up and I think that's the key takeaway that we sort of have to impress over the last 24 hours if you have a culture of permissibility if you allow people to say things and get away with things and claim things which frankly do allow and enable and facilitate a culture of racism this is where this is where we finish a 19 year old guy on Instagram being called all manner of outrageous things by anonymous trolls that's where we end up and so I think permissibility is the key word for me I mean someone who has encouraged that permissibility is of course pretty patelle she weighed in on the abuse suffered by players saying I am disgusted that England players who have given so much for our country this summer have been subject to vile racist abuse on social media it has no place in our country and I backed the police to hold those responsible accountable now this pretty patelle who is now speaking out against racism in football is the same person the same home secretary who said only a couple of weeks ago that fans have the right to boo players for taking the knee and that she doesn't support the anti-racist protests of footballers because she sees it as gesture politics now as we've shown you earlier in the show actually those players taking the knee has had a material impact on attitudes to race and racism in this country which is probably why the Tory parties weren't in favour they of course don't have any opposition to gesture politics they do it all the time that's what clapping for the NHS was while offering them a real terms pay cut that's what Boris Johnson was doing when he was standing with all those flags outside Downing Street these are people very very comfortable with gesture politics they're just uncomfortable with anti-racist politics yet they're all crawling out from under their rocks to condemn the consequences of their giving permission to racism Labour have come out and pointed out some of the hypocrisy of this Deputy Leader Angela Rainer tweeted let me be clear the Prime Minister and the Home Secretary gave licence to the racist who booed the England players and are now racially abusing England players Boris Johnson and Pretty Patel are like arsonists complaining about a fire they poured petrol on total hypocrites now I think in terms of party politics what you have seen over this tournament is the Labour Party have discovered oh look actually Gareth Southgate he's doing something which is potentially consistent with what we want to do when people like him so we're going to lean into it Keir Starmer obviously was not capable of coming up with his own inclusive version of patriotism I would imagine he would have sat on the fence when it came to taking the knee obviously he did it last summer since then he's been a bit more lukewarm when it comes to Black Lives Matter but Labour now leaning in now that Gareth Southgate has given them permission we're going to go to a couple more comments although actually the Burnley stunt was worse than all lives matter and it was white lives matter I have been informed Anne Hayfield with a £5 donation says I love you guys but I'm fed up of hearing about football is there nothing else going on in the world well I'm going to push back against that because I do think that this is you know this is a huge cultural event this competition and the England team have been absolutely extraordinary in terms of making this politically relevant so I think it is very much something we should be commenting on we have one more football story and then we're going on to coronavirus so if you are if I haven't convinced you with that comment about football just wait about 8 minutes Aaron you want to come in briefly on that yeah I mean we've obviously got a bit from Angela Reiner there but not from Keir Starmer he did do a clip people can go and watch on Twitter and what's really interesting for me Michael is that the way that Gary Neville talks about the issue succinct to the point and he's basically saying what Keir Starmer says is in his own clip but Keir Starmer kind of speaks in this really strange multi kind of syllable legalese and you're not quite sure what he's saying you need to have gone to university to understand what he's saying and you know it kind of really brings home to me the fact that somebody like Gary Neville who just speaks in the kind of idiom of everyday people he would probably be a better politician that's not to say Gary Neville would solve everybody's problems but it's really interesting for me when you have a politician who just seems so just rhetorically so distant and you have a former footballer I mean in a way of course it makes a whole world of sense who actually just gets straight to the point in terms of how they talk. Gary Neville should maybe do some media coaching for Keir Starmer Michael he's a labour support so why not that could definitely be helpful Peter Harrison tweets on the hashtag one side young POC you have the guts to step up and take pens in front of millions knowing that if they miss they're going to get a whole load of shit the other side the people giving them that shit don't even have the guts to own their vile opinions enabled by the government again nothing to disagree with there and our final football story for the evening one second here we go Marcus Rashford has been a fawn in the side of the conservatives with his campaigns to feed hungry kids yes that is something our governing party are averse to doing now that context meant that right wing pundits were quick to jump on the opportunity of Rashford missing a penalty to have a dig at the footballer one of those was Darren Grimes he tweeted honestly though Marcus Rashford penalties not politics from now on a now that was an incredibly unpleasant response from a professional twitter troll it got ratioed on social media didn't go down very well and we're not going to go through all of the reactions I just want to show you one response which is pretty indicative of the attitudes of most people to that comment from Grimes this is Navarra Media's very own James Butler responded Rashford is an internationally acclaimed athlete who has changed thousands of children's lives for the better in his spare time you're a grifting parasite reviled even by those on your own side shut your mouth ouch that tweet by the way got a lot more likes than Darren Grimes original one of course Grimes is someone whose whole career is based on creating these controversies soiling himself in public for attention we aren't going to spend a segment talking about however the reason this is relevant is because while the likes of Darren Grimes expressed this publicly there were people in positions of much greater significance who were thinking the same thing and saying it privately now one of those was conservative MPs Natalie's elfic now this is a leaked message to a Tory WhatsApp group so Natalie elfic said they lost would it be ungenerous to suggest Rashford should have spent more time perfecting his game and less time playing politics that's basically exactly what Darren Grimes was saying but she's saying it in private now the watermark you can see there is because this tweet was leaked to GB news Aaron my question for you is how much separates those sitting on the Tory benches from you know the nastiest trolls on right wing Twitter they're obviously not brave enough to say this publicly but they're saying it well this isn't even privately is it because it's to all the other Tory MPs who have been whispered under her breath yes it's clearly a spectrum isn't it look I think this is an important thing to say Michael Marcus Rashford is famous and he ends lots of money because he's a good footballer so of course that his priority should obviously be you know being a good footballer but when you're a footballer you have a lot of spare time like a lot you need time for recovery you can only train so many hours a day you're only maximum playing two games a week there's only so much football you can do right and so you have lots of time to recover and what this has been sort of historic clear is that you get lots of young men too much time too much money developing addiction problems Keith Gillespie developed a gambling problem there's one memorable example Tony Adams with alcoholism and this is a repeated theme and even people who didn't develop gambling addictions you know gambling was rife you have some of the stories for instance with the England teams in the early 2000s they owed each other from playing cards you have lots of spare time and what Marcus Rashford has decided to do with this spare time is to basically help the most vulnerable the most exploited the left behind people of this country young kids that can't get schooled dinners he's done his book to inspire young people to read he's tried to and he's been successful in changing government policy on a range of issues that have really affected low income people again particularly kids so that's how he's decided to spend his time now I wonder do Tories and Tory MPs and this lady here Natalie Elphick was it Michael do Tories like this would they rather you have a footballer like Keith Gillespie too much time too much money developing addiction problems which destroy their lives or they could be like Marcus Rashford and it should be said other footballers too by the way and he can doing this who say I have this spare time I have these resources most importantly I have a profile where actually I can change things for the better and Rashford's been very honest about this he said look I might only have 10 years of fame I mean in all likelihood that's what will happen 20 to 30 maybe a bit more if you're lucky but 10 years of fame and he wants to use it for social good and the reality is Michael sadly Tory MPs like Natalie Elphick would rather he becomes like a Tony Adams develops an alcohol addiction or a Keith Gillespie develops a gambling addiction then actually do something positive and political and it should be said what he's doing is very political Michael he's playing a very active role as a citizen in the country's politics they don't want him to do that and so this nonsense that you got from Natalie Elphick but also from Darren Grimes that he needs to focus on penalties and not you know poverty or politics it's crap what he's saying about their snobs who you know an original idea in regards to football or actually public policy or actually just helping other people has never entered their heads so Marcus Rashford is the third fastest man United player in the Premier League era to reach 50 goals after Wayne Rooney and Cristiano Ronaldo Darren Grimes still has his mum do his cleaning because he lives in our basement so you know when it comes to these points I think it's important to say Darren a few ideas should kind of stick in your brain one is stones another is glass houses anyway I want to get back to Natalie Elphick because as I say she's the most significant person here she I assume on learning that her private WhatsApp message was about to be leaked leaked put this tweet out publicly so she said last night I shared the frustration and heartbreak of millions of England fans the team gave their all congratulations and onwards to the World Cup now why I think this is a particularly pathetic tweet from this Tory MP is that first sentence last night I shared the frustration and heartbreak of millions of England fans and why I think that's just so cowardly is because she's doing what these right-wing ideological vicious conservatives like to do when they are you know essentially dismissing or abusing black players which is to say oh what I expressed you might find it objectionable but ultimately I'm just expressing the will of millions of good honest people in the country now that's just not the case here right I mean what we see from polling here is that people are overwhelmingly supportive of Gareth Southgate and what he stands for they're overwhelmingly supportive of the England football team they aren't annoyed that Marcus Rashford is standing up for hungry kids are incredibly supportive of it because the extreme opinions of Natalie Alphic which is that this incredibly articulate brave footballer should stop trying to feed hungry kids right when she says that she's not representing millions of people in this country she's representing a really vicious minority whose influence is massively over exaggerated because we have an incredibly right-wing billionaire owned press right so this idea I was just sharing the frustration and heartbreak of millions no you were just showing yourself to be the reactionary you know vicious bigger that you are that's my bit on Natalie Alphic and we're going to move on to COVID-19 before we do that we do have some news about the show we told you this before but I do want to emphasize it because it's we're so excited about it we are currently hiring another member of our production team so expanding the Tiskey team as I've said before this is because of your kind donations we're going to be building the show potentially making it more often improving the quality everything and we want your ideas to make that happen so if you are interested in the role do check out the details on our website if you think you have the qualities we are desiring then do get in touch we really do encourage you to apply the link to do so is in the description of this video let's move on to COVID-19 as we prepare to drop a new load of restrictions on the 19th of July the Tory government has been giving out a lot of mixed messages this is no more the case and on the question of masks now to jog your memory this was community secretary Robert on Sky News on the 4th of July when he was asked if he would be dumping his marks on the 19th of July well like many people I want to get away from these restrictions as quickly as I possibly can and we don't want them to stay in place for a day longer than is necessary we are going to I think now move into a period where there won't be legal restrictions the state won't be telling you what to do but you will want to exercise a degree of personal responsibility and judgement so different people will come to different conclusions on things like masks for example and the Prime Minister will set out more detail on the national policy on some of those restrictions in the coming days so that was just a week ago the following day Boris Johnson laid out plans for July the 19th they scrapped mask mandates in any and all settings now during that week I think probably interestingly actually government spoke people were asked so what if businesses want to introduce a mask mandate if they say you can't come into the supermarket without a mask a Tesco or the London Underground decided that and they said oh well they'll have to check if that's consistent with the qualities law to suggest that actually the government is not going to protect any institutions from introducing their own mask mandates unless it discriminates against people who don't want to wear masks so it was really really unsensical to say oh you should just take you know you should make your own judgement about the personal risk you are willing to take that's ridiculous because as we've been told throughout this pandemic you wear a mask not to protect yourself but to protect other people around you so the idea that you'd make a personal risk assessment doesn't make any sense whatsoever it was a stupid policy thankfully then it seems there has been a bit of a U-turn on this front seven days after Generic said master just a personal which is dependent on personal risk the vaccines minister Nadim Zahar we went on Sky News to say this the guidelines that will set out tomorrow will demonstrate that including guidelines that people are expected to wear masks in indoor enclosed spaces and of course to remain vigilant with you know hands and face and to just remember that if we all act responsibly as we did with the vaccination program the nation came together to vaccinate 80,000 vaccinated volunteers came forward at a retirement and to vaccinate we can come together and deal with this pandemic in a way that is responsible by thinking about our own actions and how they impact other people including of course people who may be immunocompromised now that message from Nadim Zahar is much better than the one from Robert Generic it happens to be the complete opposite of what he said but you know the government clearly got to a slightly better position they're saying you should wear a mask for other people they're completely abandon this idea that it's based on individual risk and the risks you are willing to take as an individual he's also brought up immunosuppressed people people whose vaccines might not have been that effective because their immune systems don't function as well as one would hope right that's sensible what's being said there so what prompted this change it's an interesting question it could be scientific advice often when the government have changed policy throughout this pandemic they've said oh the science have changed the science has changed the problem there the science hasn't changed here they've been saying the same thing the whole time which is that you need to wear a mask to protect people around you and also the idea of getting rid of mandatory masks on the 19th of July is ridiculous this was never a policy which was endorsed by scientists as you saw actually in press conferences last week where Chris Whitty and Patrick Valance stood next to Boris Johnson and said they'll still be wearing masks in public places in indoor public places what then does explain the shift now it might not come as a surprise to you that it seems to be that the Tories got cold feet when they found out the public weren't particularly keen on the policies they'd just put forward essentially they were worried it was going to damage them electorally I want to show you some interesting polling which I think essentially explains this change of tune it's from YouGov they asked members of the public whether they thought face masks should continue to be mandatory on public transport after the 19th of July 71% said yes they agreed it should be mandatory only 21% thought it should no longer continue to be mandatory so the Tories by saying that all of all of these mask mandates will end they found themselves radically out of step with the public we can also see here in shops so when it comes to shops 66% of the public thought that masks should continue to be mandatory in shops only 27% thought that should be dropped that masks should not be mandatory in shops worth saying actually the government is still out of step with the majority of the public here because they're still saying it's voluntary they're now encouraging it instead of saying oh it's up to you but what the public want and what the scientists essentially want as well is to say look let's just keep masks in places such as the tube and in places such as shops there's literally no reason to get rid of them whatsoever I think personally it's going to reduce freedoms if you have people not wearing masks in shops because they'll say no we can't have government putting down laying down these dictats because people need to be able to make their own choices I think what's obviously the case here is that forcing people to wear masks on the tube really really you know it's not meaningful freedom to be able to not wear a mask on the tube I mean it's like I don't want to downplay it's a slightly meaningful freedom to be able to go on the tube and not wear a mask but more important is to be able to go on the tube and not worry about getting Covid 19 because someone is not wearing a mask right so I think net freedoms definitely increased by people wearing masks on tubes the government didn't think so they seem to be coming around to that but only because the public disagreed for me this has real echoes of you know the whole I mean the shit show essentially which was the government's herd immunity policy they said look the sensible thing to do here is to let everyone get the disease to let it run riot it turned out the public didn't like that exactly the same things happened here what Sajid Javed wanted to do is say I'm now the health secretary for Tory back benches that means I'm going to get the public used to Covid 19 we're going to have a policy where we let it run riot so that we can move on the public have spoken today unsurprisingly don't want that people don't like the idea of encouraging a disease which we still don't know that much about to circulate in an unrestrained uncontrolled way I want to move on to today because in Westminster we got further news about what to expect after the 19th of July Sajid Javed spoke to MPs he introduced what those policies would be from that day onwards and he said this Mr Speaker as we make these changes it's so important that people act with caution and with personal responsibility for example everyone should return to work gradually if they're currently working from home they should try to meet people outside where that's possible and is expected and recommended that people should wear face coverings unless they're exempt in crowded indoor settings like public transport so he's saying there instead of last week when he was talking about putting Covid behind us everyone needs to go back to work we should learn to live with it he's now saying actually you should take some precautions even if the legal demands are falling away you need to take social, personal measures to try and limit infection this is a significant change in tone it's a welcome one even if it wasn't a change which was done for particularly impressive reasons they were worried they were going to get punished at the ballot box it does however so this policy of saying we're going to get rid of all mandatory measures but we're going to ask you to use your own judgment this does leave a lot of ambiguities what are we supposed to do the nightclub is open but am I supposed to go to it I don't have to wear my mask in the supermarket so if I find it really annoying am I allowed to say look some people don't really wear it mind wearing the mask but for me I find it particularly annoying are you allowed to not wear it how do we make this assessment that we're all supposed to do what is the common sense in the situation which will be in place from the 19th of July well all of these questions I think were were put very well by Chris Smith from the Times in today's Downing Street press conference let's go to that I think what we want to do is get people to think carefully about the ending of the restrictions that were announcing today and to make sure on the immortal words of JVT Jonathan Bantam they don't rip the pants out of it or tear the pants out of it I think is what he said ages ago and I think I said last week don't be de-mob happy about this this is not the end of Covid it requires constant vigilance it means thinking about others as well as yourself it means thinking about wearing a face covering in confined spaces where you meet other people that you don't normally meet as I've said as we said earlier on and we said last week it means continuing to think about the risk of transmission to the shielded the extremely vulnerable how you behave with them in their presence obviously exercising common sense because the legal restrictions have come off should not be taken as an invitation by everybody simply to have a great jubilee and freedom from any kind of caution or restraint that's what we're saying I think what the scientists are saying is this is the right date or as good as any other date to do this but it's got to be taken seriously and we've got to go as slowly as we as we can given the constraints we're under and I think that for me that makes a great deal of sense if we held off till September if we waited to go for the legal unlockings that we are then we'd be doing it as I said in a context of a great deal more risk from the weather plus we wouldn't have the advantage of the school holidays and so on this is about as good a time as any but to do it properly and sensibly everybody has got to be cautious so my apologies we actually included the wrong part of Christmas question the question that we saw Boris Johnson answering was much more about those ambiguities what should people do the club is open should they go to it you saw there Boris Johnson's answer he also interestingly went on to this point which there does seem to be agreement about from most scientists who are close to the government which is that the decision in terms of releasing restrictions now isn't whether or not we will have a rise in hospitalizations or none at all it's whether we will have an exit curve as we leave the lockdown now or if we do it in September the advantages of doing it in September will be more people double vaccinated the advantages of doing it now is that it's in summer so it doesn't coincide with people in schools and it doesn't coincide with the beginning of winter so real it's a difficult one in terms of should we be loosening restrictions at all but this isn't one of those situations where the government is doing something which is obviously terrible they are when it comes to mass and things like that when it comes to should we have any loosening of restrictions I think less so I do want to go back to that question how people should behave though after the 19th of July it's politically interesting it's also practically useful for us to all have an idea of how we should be making decisions after the 19th of July unsurprisingly the answers from Chris Witte and Patrick Valance were more constructive than those from Boris Johnson so let's take a look at their answer to those questions I think don't think that the things which are going to make a difference will come as any surprise to any in the UK over the next period the things that we've all been doing for a long time people have been incredibly good and incredibly patient of doing it the hands-face-based thing wash your hands cover your face in crowded areas avoid crowded areas where you can avoid unnecessary meetings but be sensible on it and above all the thing which is new get yourself vaccinated it protects yourself it protects your family it protects everyone around you these are the things that people have done very steadily and as the Prime Minister said the fact that we can move from working at home doesn't mean that we should rush at it everything should be done steadily and it's really a message about going steadily and that's what people have done all the way through this they've been incredibly good at it actually all the way through every you know across the country and all four nations and that's all we're really saying is carry on being steady can I just add two to that one is the absolute necessity to continue self-isolating if you test positive very important to stop spread and and if you're symptomatic get a test and in reopening make sure that ventilation get to get to priority so summer is quite a good time to do that obviously make sure windows are open and there is sufficient ventilation in buildings that was very good advice you know Chris Whitty essentially saying the most important thing is here if you are sick get tested and if you do have covid self-isolate I think that's something that actually hasn't been emphasised enough by the government because I think a lot of people are now thinking if I'm in contact with someone with covid-19 they're telling me it's not the most dangerous disease ever do I really have to take ten minutes out of my life because I might possibly have it I mean the messages here are getting ambiguous and I think understandably it's reasonable that we might think that self-isolation for ten days shouldn't be used in all circumstances now given that it's a less deadly disease but it's really important people don't start seeing this as just like the common cold where if you've got covid-19 but you feel okay you can go out and about also ventilation something that Boris Johnson has never properly emphasised but Patrick Valence clearly recognises the importance of I just want to bring up one final change and before going to Aaron on his thoughts on these new announcements which is on vaccine passports so you will have known from previous shows that a review led by Michael Gove found that vaccine passports would be unnecessary so a vaccine passport this is the relevant point here is not for going abroad we know that they're going to be introduced it's more for going into big events night clubs etc will you have to prove that you've been vaccinated to get in there now the review from Michael Gove found that that wouldn't be necessary during the summer it seems to have been a bit of a U-turn there obviously that review was before the delta variant started spreading today to be announced that venues will be supported and encouraged by the government to use the NHS app to check the vaccination status of attendees so it's not going to be mandatory but clubs will be encouraged to do it so it's quite likely that if people try and go if you try and go to a night club this summer you might be asked whether or not you've been vaccinated Aaron I want to go to your general thoughts on the announcement today I suppose especially what you think about this U-turn on vaccine passports yeah the vaccine the vaccine passport thing is interesting isn't it I mean it's just sort of creeping in I think you're right Michael I mean it's important to say I do agree with the ending of restrictions I will still however be wearing a mask and in closed spaces I think going into even if they weren't mandatory for instance in pubs and restaurants as you go in I don't see really the harm or the downside isn't doing that I think perhaps even people might in the longer term this might sort of crescent of a cultural reset during the winter months if you don't want to get flu cold you go on the tube even if Covid-19 which isn't going to happen completely disappeared I think that would be a smart way to behave so it is kind of it is an interesting one and I don't quite understand maybe you can explain this to me Michael what's the obsession with not wearing masks it doesn't seem a particularly onerous thing to have to do you know you cannot wear a mask on the bus or the train for this to be like the generating guiding idea behind your politics I don't really grasp it you can by the way if you're on the train you can eat you can drink right and you take your mask off and then you put it back on again or if you have a health issue you don't have to wear a mask what's the big problem Michael in terms of what sections of the population see it as a big problem not many people do the overwhelming majority of the public are happy to have a mask on public transport and would prefer other people to do so but the people who have an ideological opposition to mask I think you know it's kind of cranky people it's people who never thought Covid was a big deal and so discover their libertarianism when it comes to these arbitrary restrictions even though they're perfectly happy to have a very strong state when it comes to crime and punishment or things that would never affect them they're basically affronted that they have had any intervention which they have to follow equally they don't like that kind of intervention when it comes to what Sajid Javed was thinking in terms of getting rid of these masks my theory here is that Sajid Javed entered that role and his key aim was to say we need to get the public used to getting Covid-19 we need to get the public to have an attitude where one cannot really avoid getting it so let's go back to normal let's treat Covid-19 like the common cold so he wanted to as much as possible get rid of all the visual signifiers that were in the middle of a pandemic and obviously one of the key visual signifiers that were in a pandemic is the mask so he wanted to get rid of the mask and say everyone pandemics over forget about Covid-19 if you get it you get it think about it like a cold and that's why he was so obsessed with removing that mask mandate what's happened since then is that the scientists have come and said look if you if you don't even have masks there's gonna be an absolute nightmare the public have also said we don't care about masks this just seems wildly responsible to us Sajid Javed has basically realised that you can't at this point persuade the public the pandemic is over because it's not and the public aren't actually very ignorant about Covid-19 we know a lot about it now you know the vast majority of the public know a lot about Covid-19 you can't suddenly click your fingers and lie to them and say the pandemic's over so they've done a U-turn on that policy the idea that we can snap our fingers and suddenly no one cares about Covid-19 that policies over that's why it's a little bit like that U-turn we saw in March 2020 where the government's policy was to say we need everyone to relax and get used to the fact that some of their families are gonna die some of their family members are gonna die and to just move on and take Covid on the chin and then they discovered that's impossible so they had to do a complete U-turn the same thing is happening here they tried to do herd immunity mark 2 in a sort of forget about the virus let's just let it happen people didn't like it they should have learnt by now I don't really know why they thought that was ever plausible but to me that's why that U-turn has happened but Michael I mean that was a policy choice it was a ridiculous policy choice herd immunity but it does feel with Sacha Javid on this and the mask obsession that it is purely and you're saying it's a signifier about this thing being over I think I agree it's a signifier but for me it's just a signifier of this like abstract near liberal idea of freedom I think even I don't really exist they're incredibly cheap they're quite effective like you say people like them I don't really they don't really they have no impact on the economy I don't I don't really unless it is just this abstract sort of this is a guy by the way I'm sure our viewers already know this you know but you know he's such Javid reads you know I ran to his wife every year you know this is somebody who is an intellectual ideological neoliberal they don't think the state can do anything and I think you know in a way you know because by the way we have lots of constraints in life right if you go on the train and you go not wearing underpants you're gonna get done for public indecency there are lots of constraints in life and I think there is a part of his sort of politics is Randy and politics quite frankly he would quite like you to be able to sort of go on a train and you know not have to do anything you can say what you like do what you like and I think for me Michael and you because we're coming out of this we forget this but this pandemic the last 18 months has just been a sledgehammer so that whole idea of politics of sort of Randy and ultra libertarian politics and because we're coming out of it now I think we could really talk about that a bit more and I think we should be able to look back on that the last 45 years of this whole ethics and politics and way of being and say what the hell was that about not only does it not work not only does it not make sense but it ends in obsessions about the most ridiculous and strange things like not wearing masks after in this country 130,000 people have died the death knell of neoliberalism the death knell of iron round that's what we should be saying Michael as we come out of this we go what the hell were you thinking for the last 40 years yeah I mean I think I think that's the Julia Hartley brewer position here I do think that Sadid Javid it's more cynical in a way I think it's less honest because I think essentially his priority is we need to get the economy back to normal back to how it was before because we don't want to normalise government intervention so he wants us all to flood into the bars so that no more support is needed flood into the nightclub so that no more support is needed flood onto the high streets so that no more support is needed he thinks that so long as we think there's a pandemic going on we're not going to do that I think his worry was there'd be some cognitive dissonance if you went on the tube with a mask and then you took off your mask to go to a restaurant or a bar people say well if I'm not safe on the tube why am I safe in the bar he wants to get those numbers back to normal so his theory was well we'll just take masks away from everywhere the problem was he discovered that actually that wasn't going to get people to flood into the bars because they thought the pandemic was over that was just going to mean that people wouldn't even go on the tube so they completely misunderstood where the center ground of public opinion was on this one and they've U-turned because of it again as I say that that's why it's a bit like that herd immunity U-turn that we saw in March 2020 so you think policy I just think it's pure randian ideology nonsense but you know we'll have a bit of a nuanced disagreement on that I think yeah I think we can I don't think that's going to be a clincher there's not going to be a split in Navarra because I think it's for the economy and you think it's because of a randien obsession let's go to a couple of comments or first of all thanks to Mary Dwyer who donated a 10 pound super chat and then Rob Hogg with a tenor who says even if your only priority is the economy masks still make sense since that bozo blabbering of freedom our local hardware shop has had to close and an eatery cancel an event because of COVID that's something that's been said and throughout the pandemic I suppose to play devil's advocate or to slightly defend the government's position when it comes to the fact that COVID cases are going to increase they're saying if we allow COVID cases to increase now then we're going to get what essentially some form of herd immunity both by vaccinations and top ups of infections if we really suppress COVID now then we'll just have those same peaks and that calf will have to close in September you know in a way that's beyond my pay grade to quite understand or explain to you why that's an inevitability but this does seem to be something which the scientists are very confident in at the moment there will be an exit wave and we are making a decision about whether that will be in July or whether that will be in September let's end the show for now Aaron Bostani it's been a pleasure as always it has been my pleasure Michael I just want to say by the way because I missed the start of the show it gave me an even greater appreciation for what you do Michael the fact that I had to host a show last week you are a consummate professional you're only getting better and I think it's fantastic we're going to five days a week and you're going to be you're going to be hosting almost all of them which is obviously a huge ask of you but I think you're ready to step up Michael and that's what the last week taught me I very much appreciate those comments I have to say my one regret today is that I was hoping that I'd come back from my week away incredibly refreshed I've actually come back from my week away incredibly hungover because of England making it to those finals it would have been worse if they'd won I have to say for now thank you for everyone for your super chats tonight if you want to support us directly then you can go to navaramedia.com I'll be back on Wednesday at 7pm so make sure you hit subscribe and tomorrow Ash Sarko will be talking to Juliet Jakes on how art can fight transphobia that will be at 7pm if you're watching on YouTube you'll be taken over to that stream when this show ends for now you've been watching Tiskey Sour on Navarmedia good night


Welcome back. Today, we will discuss sums of random variables. Last class, we discussed maximum and minimum. So, given n random variables, next went through x n. We figured out how to, given their joint series, we figured out how to get the distribution of the minimum and maximum. Today, we will discuss sums, sum over sum of x i's. So, first of all, so let us say that, let us just take 2 random variables. Let us say x y random variables on omega f p. So, you have 2 random variables x and y on omega f p. Just for you to remember the picture, you have omega and your x comma y is a map, is a map from omega to r 2. And given any borel set on r 2, the pre measures are f measurable, we know that. Now, the question is, if I take, so if I define z of omega equal to x of omega equal plus y of omega. So, for every omega x of omega comma y of omega is a pair of real numbers, it is a point in r 2. I am interested in x plus y, which means, I am looking at the random variable z equal to x plus y, which means for every omega, I have z of omega equal to x of omega plus y of omega. And I want to ask, what are the statistical properties of z? In particular, given the probability law or joint CDF, joint CDF of x and y on this, on r 2, I want to figure out, what the CDF of z is, that is the question at hand. However, before you go that far, you have to convince yourself that z is in factor random variable. What do I mean by that? z must be a measurable function, z must be an f measurable function. We know that x and y are f measurable functions, but we do not know if z is always a measurable function. It turns out this requires a proof. So, the proof is actually fairly simple. It exploits the fact that, it uses a fact from real analysis. It uses the fact that, the rationals are a dense countable dense subset of r. So, let me write this down. So, z is indeed a random variable on omega f. In other words, I have to show that. So, I have to show that, sets of this form, let us say I have omega such that z of omega greater than little z. So, I have to show that, these sets are f measurable or equivalently less than or equal to it does not matter.  I only know that, x and y are random variables. So, what I can do is, I can write using the fact that, rational numbers are a dense countable subset of r.  So, a dense subset of, let us say a dense subset of a metric space is a subset such that, the limit points of that set is the whole space itself. So, the set, if you collect the set, if you take the rationals, the set of all limit points of the rationals is, in fact, the whole real line. Every real number can be written as a limit of a sequence of rationals. So, which means that, rationals are dense in r. So, using this fact, we can write as follows. We can write, this is union q in q such that, omega for which x bigger than q and y bigger than z minus q, which is the same as saying, union q in q, x greater than q, union y greater than z minus q. And the union is over the whole thing. So, whenever, what I am saying here is, whenever x plus y is greater than z, strictly greater than z, there always exist some rational number q. See, there always exist a real number q such that, there always exist some real number such that, x greater than that real number and y greater than z minus that real number. But, since in the vicinity of any real number, I can find a rational number. I can also find a rational number, which satisfies that. Now, why am I so worried about a rational number? Because, I want to induce a countable union. So, I want to write this as a countable union of these guys. Now, if you look at this, so I have just written this as well, I think I mess this up. I think this should be an intersection, is it not? So, this be an intersection? Yes, this should be an intersection. So, this is f measurable, why? x is a random variable that is f measurable. So, because y is a random variable.  So, sum of two random variables is a random variable under no further conditions. It is always a random variable. Similarly, you can show that product of two random variables is a random variable, similar argument. Anyway, so we are not discussing products now, we are just discussing sums. Is this argument clear? It uses the fact that, the rational are dense in R. So, even just looking at this equation, you can easily figure out the CDF of Z. This expression itself gives you a, in principle a way to figure out the CDF of Z. Because, you can write probability of that, which will be the 1 minus the CDF is equal to some probability of the countable union of a bunch of this is a complementary series after all. So, you can use some probability properties to eventually figure out this. In principle, there is no problem. However, so in the special case, we will in the special cases, cases where x and y are discrete and in the case where x and y is jointly continuous, we will derive more specific formula. We have the expression simplifies a lot more. So, for everybody with me. So, if you are looking at, so you are basically looking at the probability that, let us say this is the probability you want. This is Z is equal to x plus y. So, if you are looking at, let us say you are given a CDF. So, let us say this is my x y plane and my CDF is something on this. See, joint CDF is given on this, joint CDF x and y is given. So, it is start of minus infinity. It will increase and go to, it is at minus infinity start of it is 0 and go to 1. It will be some three dimensional surface, three dimensional figure on this. Now, what I want is the probability that Z is less than or equal to little Z. So, I draw the line x plus y is equal to little Z and I am looking at the total probability mass sitting on this side of this line below the line. So, that is what you have to figure out. So, you have to look at the joint CDF and figure out how much of the probability lies below this line. If you in general this may the joint law may consist of some masses, may consist of some densities, it may consist of some crazy stuff like canter function or whatever. It can have all sorts of things lying here. You have to somehow add up those probabilities. In the case when x and y are both discrete, the job simplifies a lot more. That is because of x and y are both discrete, you will have only, it will take only a countable set of values. And for any given Z, you add up all the masses sitting on this side of the line. In that case that should be easy. So, the discrete case is very easy. And in the jointly continuous case is also very easy, because you have a joint PDF, which you can integrate over the set. So, those two cases we will cover. So, let us say that. So, this is what you want. Let us say first assume. So, this is what you want. Assume x and y are discrete. And if x and y are discrete, I know that my joint PMF is the complete specification of these two random variables with given joint PMF P x comma y. Then probability that. So, if I want f z of little z is equal to, I will simply sum over all the x's and y's, such that x plus y's are equal to z. So, I will simply write this as. So, I sum over all x and y's, such that x plus y is less than or equal to z P x comma y of x comma y. So, this I can again write as sum over P x y of x comma wait. So, let me. So, this is joint CDF. So, actually let me write this. This is easier to get just a minute. So, for the discrete case, I should just do this, because z after all will also be discrete, because x takes only countable values, y takes only countable values, x plus y will take only countable values. So, I can actually go for the PMF. So, I can instead of writing down the CDF, let me just do this. So, here it will be this will become x plus y equal to z and this will become. So, there is nothing wrong with what I was writing earlier, but this is simpler. So, this I will sum over all x. So, I am summing over all x y such that x plus y equal to z. So, I may just put y equal to z minus x and sum this PMF. And if you know the PMF of z, you will you can get the PDF P or CDF. So, you are summing x out. So, this will not depend on x, it will only depend on z, you get your PMF. And so, this is in general, this is the general formula. In particular, if x and y are independent random variables, then I can simplify this further. I can write P z of little z is equal to sum over all x. So, this will become P. So, this will product out into the marginal PMF.  This is only true of x and y are independent. So, this is like a discrete convolution of the two PMF. You would have seen the convolutions in your signals and systems. So, this is a. So, this is denoted by P x star P y. So, discrete convolution of the individual PMF or P y convolution P x. It is the same thing, F convolution G is same as G convolution F. So, if you are given two independent random variables and you want to find discrete random variables. And if you want to find the PMF of the sum, you will just go ahead and convolve the individual PMF. But only if they are independent, otherwise you have to do this. This is the more general formula. So far, let us do an example. Let x 1 and x 2 be Poisson random variables independent Poisson random variables with parameters lambda 1 and lambda 2. And let us say you write z is equal to x 1 plus x 2. Then P z, z is given by P x 1 convolution P x 2 of z. You know these marginal PMF because then given to be Poisson. So, P x i of x will be equal to e power minus lambda i lambda i. Let me call it k, because it is just a integer value random variable P x i of k over k factorial. And this is for k greater than or equal to 0 actually k in integers. Now, let me try and do this convolution.  Now, this looks ugly as of now, but you can simplify it. So, what you will do? So, this looks invitingly like z choose k. So, what shall I if I put a z factorial here and divide by z factorial, I will get a nice term. So, let me write this as e power minus lambda 1 plus lambda 2.  Did I leave out anything? I think I am fine. So, that is sub binomial sum is it not? So, this looks like lambda 1 plus lambda 2 power z. This is the binomial expansion of. So, z is all. So, I am just looking at z in integers.  So, if you take 2 person independent person random variables, the sum is also a person random variable whose parameter is lambda 1 plus lambda 2. In fact, you can show that this holds for even some of n independent person random variables with parameters lambda i. The sum will be also pose on with parameter some of those lambdas. So, this holds for z in 0 1 2 dot. These are all for this is distribution over integers non negative integers correct. This is an element of z plus. So, let me just write let me just write 0 1 2 dot dot dot whole numbers. Any questions? Next we will consider the case where x and y are jointly continuous. So, the discrete case we are done we are going to go consider the jointly continuous case. These 2 special cases you get nice expressions.  So, if you do not have if you have some crazy random variables x y with some very bizarre CDF, you have to simply look at the probability mass below this line. There is no simple expression for it, but in the case when x and y are discrete or x and y are jointly continuous, you will get some nice formula. And these are the cases of maximum applicability in engineering. So, these expressions are useful to know and useful in practice as well. So, let us say x and y. So, we finish the k assume now assume next that x and y are jointly continuous. If they jointly continuous I have a joint density joint pdf with joint pdf f x y which is given to you. Now, I want the density of. So, I actually want let me say I start out by saying I just want the CDF of z right. What you will see is that z will also actually be a continuous random variable. So, not only so we can actually get its pdf as well right. So, this is what we want.  So, this is probability of big z less than or equal to little z. This is simply. So, you know that for any. So, you know that the probability of any borel set is given by the integral of the joint pdf. So, I am now seeking out. So, this is what I want the probability of this set is what I want right. The set that lies below the line and that is a borel set you can show. So, the probability of lying in that borel set will be the integral over that set of little f x y right. So, it will just be integral x y such that x plus y is not equal to z f x y of x y d y d x or d x d y does not matter right. This you agree this is rate on actually right. So, now this integral. So, after also if you write this as a double integral on r 2 this will become integral minus infinity integral minus infinity z minus x f x y of x y d y d x. So, what I am doing is I am first integrating y out. So, I am doing. So, if you look at this figure I am integrating from y from minus infinity to z minus x right. I am integrating along the y first and then integrating from x from minus infinity to infinity right that is what this is right this integral is. So, that is it actually I mean if this is all there is. So, you know the joint density you integrate it this is an integral you can do right and what you will be left with. So, if you do the first integral you will be left with something that is a function of z and x and then x will also get out when you integrate the second time then you will be left with some function of z. So, this is all there is, but it turns out this can be simplified a little if you put. So, conceptually it is over, but if you put let me do this right if you put t equal to x plus y then this simplifies a little bit and gives you. So, minus infinity z well let me just minus infinity infinity integral minus infinity z. So, I am just now I have put I have gotten rid of z minus x is I have put z minus x is equal to well I have put x I have put t equal to x plus y which means my y is equal to t minus x right y is equal to t minus x. So, I will have f x comma y of x comma t minus x right y is equal to t minus x and I have d y is simply d t now right. So, I will have d x d t is that fine d t d x I think. So, I am integrating this the x integral is not it yeah. So, d t d x correct this is just a substitution in the integral good. So, now this guy is non negative all right. So, I can in fact there is no harm in inter changing the order of integration that is always valid if you have a non negative argument. So, this you can write as integral minus infinity z integral minus infinity this is also these two are equal integral f x comma y of x comma t minus x this guy now I think. So, now my variable is t is not it. So, when my so I have gotten it of y right. So, y was going from minus infinity to z minus x now my t is going from minus infinity to z I think that make sense right. So, I have gotten it of y. So, this is now I have d x d t I have just flip the two integrals which I can do because my integrand is non negative. So, why did I do that if you look at this integral. So, if you look at this bit. So, that is getting rid of x right I am integrating x out. So, what remains will be a function of t right. So, I will have something that looks like a density right. So, this is this will be some non negative measurable function of t and I am saying that the c d f can be written as integral minus infinity to z some function of t d t which means this guy must be the density of z right. So, this guy must be the density of z which automatically proves that z must be a continuous random variable correct. So, which means if you give me the joint density I can easily find the density of z by evaluating this one integral right. I do not have to the integral outside simply getting the c d f from the p d f right. So, I can write thus.  So, you give me the joint density I will find the density of the sum z this is the formula for that and it is always a continuous random variable if x and y are jointly continuous z is always continuous random variable. And then you say in particular if x and y are independent you will get f z of little z is equal to integral minus infinity infinity I know that almost everywhere my density will joint density will product out into product of the marginal densities. So, I will get f x of x f y of z minus x d x this is a convolution also right this is a convolution of. So, this is written as. So, you can write this as f z is equal to f x convolution f y right. So, if x and y are independent jointly continuous random variables then you get this formula. Now, you know that if I see x and y are independent and marginally continuous it is automatically true that they are. So, jointly continuous right. So, actually. So, if x and y are independent random variables with densities x and y right you can always use this expression is this clear. So, if you have good practice in convolution you can do a lot of examples like this right you take if x is some exponential and y is some independent Gaussian you can convolve exponential and Gaussian and to say what the sum will look like right it will not be pretty right. But, some cases you get nice answers right. So, once such example you can show as if you take x as a Gaussian random variable with parameter mu 1 and sigma 1 and x y is an independent Gaussian parameter mu 1 and mu 2 and sigma 2 then the sum will have parameters some will also be Gaussian will have and it will have parameters mu 1 plus mu 2 and the sigma will be square root of sigma 1 square plus sigma 2 square. You can show it explicitly by doing this convolution it will not be pretty in the beginning, but you can figure it out right you can convolve and figure it out. So, that is one thing you can do as a home work see eventually that is not how we will do it. So, you know that convolution is not it is not cool right it is very ugly right. So, normally what do people do when you want to convolve two things you take your favorite transform Laplace or Fourier multiply the transforms right and invert back. So, that is what we will do eventually, but for now we have not gone there. So, you can convolve for now right. So, you so if you try the Gaussian example with different parameters it will probably take you about half an hour to 40 minutes to actually do the integration. But of course, if you use transforms it is a lot easier right which we will get to later. So, let me do shall I do an example I will do an example. So, let us say x 1 and x 2 are independent exponential random variables with parameters lambda 1. So, let us say mu 1 and mu 2 and z is equal to x 1 plus x 2. Now, f z of little z is given by integral 0 to. So, this p d f is known right. So, f x i of little x will be mu i e power minus mu i x right.  So, and for this you will take integral 0 to z because that is where both that is where both densities are non 0.  So, which is equal to mu 1 mu 2 e power minus mu 2 z right mu 2 z times integral 0 to z e power mu 2 minus mu 1 times x d x fine just doing the convolution. So, in this case if. So, now you have 2 cases right it may be that mu 1 and mu 2 are equal right in this case this whole thing will become e power 0 which is 1. So, the integral will simply be integral 1 d x will be z right. So, in that you will get mu square z e power minus mu z right if mu 1 mu 2 are equal to mu and if they are different this you have to integrate. So, you get 2 different forms. So, this will be equal to mu mu 1 mu 2 by mu 2 minus mu 1 e power minus mu 1 z minus e power minus mu 2 z all that this is true if mu 1 is not equal to mu 2 and this will be equal to mu square z e power minus mu z if mu 1 equal to mu 2 equal to some mu. So, if you had the parameters were the same then this will be the pdf will look like z e power minus z with mu equal to 1 right. So, this particular this is called Erlang Erlang of second order second order Erlang Erlang 2. So, if you sum 2 independent and identically distributed exponential random variables you get a what is known as an Erlang 2 distribution. And if you do this n times you will get an Erlang n distribution that will look like mu power n z power n minus 1 e power minus mu z I think there is an n factorial or n minus 1 factorial in the denominator. And the case when the parameters are different you get something right it is not some well known this is not some well known distribution this is a class this is a one of the classical well study distributions. So, Erlang has lots of applications in calculating switch blocking probabilities in telephony you know call blocking probabilities in telephony good. So, any questions on this. So, the other example. So, the homework example I was saying was if x 1 has Gaussian with parameters mu and mu 1 and sigma 1 square if we wrote down the expression for this right e power minus x minus mu whole square over 2 sigma square and so on. So, and x 2 independent with parameters n mu 2 sigma 2 square and their independent then x 1 plus x 2 is also Gaussian with parameters mu 1 plus mu 2 and sigma 1 square plus sigma 2 square. Now, I am writing squares here no yeah. So, your sigma will be the square root of sigma 1 square plus square square. So, I am writing the what is what we will interpret as the variance not the standard deviation good. So, this is this you will show actually now yeah if you try and show it now it will be a very ugly exercise, but you should probably do it just to convince yourself that when you do it in a easier way later with transforms you will have a much easier time. So, here this is a very fundamental property of the Gaussian distribution this is one of the properties it is one of the two properties that makes the Gaussian distribution in some sense the queen of all these continuous random variables right. So, if you add any number of independent Gaussian random variables you get another Gaussian random variables and the parameters also simply add. So, if even if you add n of them independent random variables you will get another Gaussian. So, it is a stable distribution under addition. So, in the discrete world you saw Poisson was stable also right. So, the continuous world this is one example of a some stable distribution this is a stable under summation. So, that is one of the reasons why this Gaussian is. So, important the other reason is central limit theorem which means that which says that if you keep adding a lot of independent random variables which are not even Gaussian you will the some will look approximately like a Gaussian. So, which this is a central limit theorem is something you will do towards the very end of the class even in proper detail before that we have a lot of ground to cover. So, this I have not proven because it is not because it is difficult because it is messy and we will do it in a more elegant way later any questions. So, if there are no more questions I will stop here.


As a demonstration of atmospheric pressure, we place some water inside a large metal can. We've heated the water up to the boiling point and the steam has displaced all the air in the can. We'll now tightly cap the can. The steam inside the can will condense. This will produce a very low pressure inside the can. There's atmospheric pressure on the outside of the can. We'll add some cold water on top of the can. The can collapses due to the large difference in pressure.


Hi there and welcome to my channel. My name is Tracy and I am a small-scale gardener living on the prairies here in Saskatchewan, Canada. So if this is your first time joining me on my channel, I enjoy making videos on cold climate gardening, how to grow indoors during long cold winters, along with sharing recipe ideas made with ingredients grown in my garden or locally sourced. So in today's video, I'm going to show you how to make a cooling herbal body and facial spray using fresh herbs from my garden. So here on the prairies, we're about halfway through our growing season, so I have been busy collecting herbs, flowers, weeds that I enjoy using for cooking, storing over the winter, and also to make a variety of different natural body care products. So making hydrosols from your garden plants is very easy to do and it has a lot of different uses that I will go through with you later on in this video. So let's head into my kitchen and I will show you how to steam distill your herbs, flowers, and plants to make a hydrosol. Okay, so I'm just going to run you through all the equipment you need to set up your very own steam distiller. It should be easy to put together with things that you have around in your house. So first of all, you just need a deep pot and a lid that fits on securely. You'll be inverting the lid this way. A clear one's kind of nice because then you can actually see the steam distilling process happening. So you need one of those. You need a couple dishes that are oven safe. Make sure that they can withstand some heat. So I'm going to be using this little ramekin as the base and something that's got quite a wide opening in here because this is going to catch the steam from the lid and that will sit on top. So just make sure it's something that can sit securely so you'll be putting it inside the pot like this. So like that and all your herbs and water are going to go around this area. It's best if you're using distilled water and of course whichever herbs, flowers, buds, leaves, whatever kind of your favorite plant that you want to use in this process, have those ready. They can be dried or they can be freshly picked right out of the garden. It doesn't really matter. And the last thing you need to have is a good supply of ice.  So if you can, if you have an ice maker of your own, of course these ice packs work really well if you have a good supply of them. And if you're planning a day ahead or so, make yourself some ice blocks or in small containers or just take some water bottles, fill them up with water and freeze them overnight and these will work good as well. Okay, so we're ready to get this set up on the stove. Okay, so once you have your inverted container or dish here in the pot, the next thing you want to do is add all your herbs, flowers, and whatever you are going to be distilling. Just sprinkle them around. For my recipe here today, I'm working on making a cooling facial spray. So I've picked from my garden some fresh rosemary, lemon balm, peppermint, eucalyptus, and I'm not able to grow lavender here very successfully. So I'm going to cheat and add a little bit of a bottom lavender buds. So then next you're going to add your distilled water. And I just pour enough in here to just slightly cover the herbs about two or three inches to start with. I'm going to be adding more water throughout the distilling process as it slowly cooks away. I usually top it up two or three times, try to get the most I can out of all these herbs. So there's about three inches of distilled water in there. So you always want to make sure that everything that you're using is sterile because you don't want to have any bacteria ending up in your product after you've created it. So I always give everything a good spray with some rubbing alcohol. Sometimes I use hydrosols and lotion products that I make.  So then you put your bowl on top here that's going to catch all the steam and then put your lid on, invert it. And to start with I turn on the heat at a higher temperature to get the water heated up quicker and once it starts to bubble turn it down to the lowest setting so that you have it simmering very gently. And as it simmers away you'll start to see the steam building up on the lid and just to speed up that condensation process. That's when you add your ice. Then I just let it simmer away for half an hour to an hour. You don't have to stand and watch it non-stop. You can go about your business around the kitchen and the house and just check on it just to make sure that the water hasn't all boiled away. You will also want to have some kind of jar or container to store your hydrosol in after you've poured it off so make sure you have something sterilized and ready. Okay so as you can see here the water is slowly steeping and steaming all these plants carrying all that herbal goodness to the lid here where the steam is condensing because it's hitting this cold surface that we've created with the ice and then from there it drips down into the bowl and then you are left with some beautiful hydrosol. So about every half hour or so I like to just remove the ice and check my collection of steam here. You want to be very careful because this is very hot steam so you should be wearing some kind of a a glove while you're doing this. So I like to pour a little bit off every half hour or so. Smells so amazing. It's like having an aromatherapy thing runging in your kitchen while you're you know while you're doing this. It's just a wonderful smell to have. And then after I've done this I usually check the water level just to max mount the amount of steam distilled hydrosol. I can get out of this here. So as I said the hydrosol I'm making today is to use as a cooling body spray or facial spray. What I love about hydrosols is that their aromas are often very soft and subtle compared to using something with essential oils. And the spray leaves your skin feeling cool refreshed and provides an uplifting aromatherapy property. So depending on what herbs, flowers, weeds or even fruit or vegetables you use hydrosols or floral waters can be made and used for many things. So with this process you can create a room spray, an anti-itch spray, surface cleaner, bug spray, bath tonic, facial toner, hair rinse or even a makeup remover. So if you've made any of these or have created a hydrosol not mentioned here please let me know in the comments or if you'd like to see me make more hydrosols in future videos let me know in the comments as well. So ideally hydrosols should be stored in the refrigerator to maximize their shelf life but they can also be stored in amber or dark colored glass bottles and kept in a cool dark location. So I hope you enjoyed watching me make a cooling body mist using fresh plants from my garden and that you will try making some of your own hydrosols. I would appreciate if you give this video a like leave a comment and please subscribe and stay tuned for the next video on my channel. Thank you for watching.