But now for something completely different, let me check if my full screen slideshow works. I think it's looking fine on Twitch. So now we're gonna get an Italian and a German trying to cook pasta together. It's not the beginning of a joke. You can guess that I'm the Italian and Thomas is the German. And usually, as it came up many times, Germans apparently they break the spaghetti, which is a big plus for me for Italians. But you can get it into a smaller pot. It's okay, just exception. So in this other motivation are somewhat funny. So we tried to introduce the concept of parallel computing using kitchen, isn't it? So my task is to cook pasta for four people. And the ingredients if you want to make it to pasta for four people is 500 grams spaghetti. Hopefully you have a mother from Italy like I do. And we have some amazing ready-made tomato sauce that she made. Four liters of water, even though it would be five liters of water, but we care about the environment. And then, of course, we need some two. We need the stove, like we see here. Stove at multiple burners. You need at least one burner for warming up the water. And then, you know, you need the pot to keep the water and the spaghetti tends to steer the spaghetti. So the algorithm that I see for doing this is that you boil the water for five minutes. Add the salt quickly, and then put the spaghetti in water and then cook for eight minutes and stir every now and then. So, you know, for those who missed the joke, the metaphor, sorry, I'm not joking, metaphor. Then we're talking about here is that the chef, the cook, the person is the process, the computing process, you know, the managers, all the virus resources. And then the pot of them in computing is called the threat that, you know, in this case, I have one pot. So I can only run one thread, one computation per time linearly, serially. Then there's the kitchen, which is the node or the computer where, you know, like the laptop, for example, where all these components are stored inside. So where there's one or more CPUs with, for example, in laptops after you have four, four CPUs these days, if not more. Then there's the burner, which is the single CPU. There's some water, which is the memory and the pasta in this case is the data that we need to process. All right, so now what the main key thing is about time, making the pasta with big time, and especially specifically for this type of pasta. Every pasta is a different amount of time, you should always check it in the envelope. For this type of spaghetti, it will take eight minutes to put them properly. So now the question is, if my store in my kitchen has four orders, would I make the pasta in whatever time divided by four weeks? You know, wouldn't I parallelize the pasta? And these yellow boxes that you see there in the bottom are the quantity examples.  Can I just move my code there and suddenly run 10 times faster? Well, the answer is no. So multiple codes at once. Well, nonparallel. So, even if we're going to try to, you know, cook the pasta, you know, let's imagine that the task would be that we would need to cook four times the amount of pasta. So then, you know, I would need to have four pots, four boxes of spaghetti, four of them are the sources, these are things that we can buy.  But me, myself, the process, I don't know how to take care of more than one pot. I never learned how to take care of two pots at the same time. So unlimited. What I can only do is that I can cook one pot of pasta, wait that it's over, repeat, you know, so to cook pasta for four times four for 16 people, it will take me four times the total amount of pasta. Unless your code specifically knows how to use multiple CPUs, it will not get done fast. And it's actually not, not so easy to learn how to use multiple pots at once. At least in the competition sites because your code commonly, well, does not do multithreading initially. Unless you're doing a really different task that you're, you know, cooking some sauce that maybe you can let rest for a bit, and then taking care of the pasta, you're not. I don't know if I would actually be able to cook four pots of pasta in one. So let's assume that I go to cooking part of the school and I learned this new technique, which is the open MP, which in this case is open multipot. Realize open MP is the language is the API whenever it's the system that is used to basically have multiple process running. So now I can put pasta for, I can put four pots of pasta in one because now I learned this, which means I rewritten my code that you can actually see the four CPUs and run four processes independently. We just now have to be, you know, a little bit more careful that our process don't get into each other way. So, you know, if I'm steering one pot, I should be careful that I don't forget the other one. So now I can actually cook pasta for 16 people. It's all in the same in the deep time because it's all done in parallel. So in eight minutes, the past 16 people will be ready. Thanks to the parallelization. This is quite a trivial simple parallelization, the four pots of pasta. They don't really need to talk with each other. As long as we the process I remember that one started maybe even a few seconds later than the other. It's, it's, it's quite a simple process to scale. But now things a little bit gets, you know, a little bit more complicated. Maybe, you know, I'm not really able to manage the four pots in parallel. I could hire, you know, I could hire three more Italians to cook pasta with me. So then the, I need to basically talk with these three other Italians, they're sharing the same kitchen, of course to talk in the kitchen. I guess you can see all the dishes. Both Italians cooking, but what they feel and think company is this a good messaging. Interface, but what was MPI? Process interface. Yes, messaging process interface, but now suddenly, you know, we need to talk with other processes. And what is interesting here that, you know, when we, if we really want to scale things out, they don't even need to be in the same kitchen. They can actually be in the kitchen upstairs in some other kitchen in the same beauty. And we could just pass these messages and okay, now it's time for the pasta and everyone can throw the pasta inside. So by adding more processes for people, we can actually, you know, get more things done once again. Things in general, don't really scare winner. It's not that by hiring 1000 chefs, I will still be able to produce in synchronous way. This, whatever 1000 times 500 grams of pasta, because then the messaging, you know, the overhead and the messaging to synchronize with all the 1000 chefs will be, will be quite, you know, we will take some of the time. Some of the cooking time, some of the sort between the different parts. So I think you think about Enrico calling all the different chefs and having a short chat. Yeah, like how do we, maybe it's easy if one person broadcasts and messages to many others, but it's nice to be interactive and one has to say, hey, my water is not boiling, please everyone.  But now let's assume that, you know, I go to a computing cluster, meaning that I live in a building apartment that has 25 apartments. I can actually scale this pasta 15, 25 times because there's 25 apartments and each apartment has a store with store where there's will do my pasta time again be able to scale so that I can put the pasta in. Well, no, because yes, the cluster as, for example, 300 nodes and every node is 40 CPUs, but I can't just throw my code to the cluster and hope that now suddenly everything goes, uses the 300 nodes and whatever 340 CPUs. It's not going to happen unless specifically by things to make it. So, so for example, you know, this could be the case that I know, or that remote nodes, meaning not just my kitchen but someone else's kitchen. And I could start funding some of the pasta. So, you know, I will have a friend upstairs, for example, another very well given the call when I use your for learners. And then you have a new flat. And so, yes, I could not have a parallel box or pasta. But sometimes the kitchen upstairs might be busy. So if I'm requesting for four points from a friend upstairs actually not every now will actually most of the time pasta and so on. So then I go to the house manager, which is sir. The company is going to slur. You learn a lot about slur in the day to entry. So slur. This sir learn knows everything about the kitchen. So if I can start asking to slur, you know, is there any free kitchen with four burners that I can immediately use. So I might need to wait to learn those that actually the names of stairs in the last floor they're soon done. So if you wait 10 minutes, we can then start your for learners. You know, so again on the cluster, there are so many computer that it would be impossible for you to keep track of them to know which one is available and which one is not available. So then enters the work of managers slur, which basically does this for us to request the amount of sources you request in this case one note, my kitchen with for safety use for foreigners, you will have to wait a little bit if there's not available. So once you get it, we can start the process. You know, there is a certain. Then, you know, there are other special things and special tools that a burner is not made for them. So now, let's suppose the case of the cheese, which is great, which is a great example of architecture. The hardware is actually as parallelization. So cheese greater than that many things at once. That doesn't work for any slicing program. Of course, here the metaphor is that the cheese that is a GPU, the GPU is the graphic processing unit was somewhat similar to the CPU. But the difference is that the architecture, the actual hardware, the actual pieces of wire that are inside there, they're actually be in the realization, which means that it can work much faster than, you know, 10 CPUs or one of the CPUs in part of it. But of course, the code and the program is to be written specifically for GPUs. So for those first, for those friends who want to achieve some of that late, you might want to use the GPU, because if I don't have a GPU, I will need to break the computer or a job to spend the life with a CPU knife. And even take me ages to break the local computer. Now the cheese creator, the GPU greater, you know, suddenly I can write the full on the job log and get all the cheese there. So a GPU can do very many things at once, but only very specialized on the job. You need specific programming skills for specific problems to use. You probably need to adapt your cheese a bit to the cheese straighter if you have one of those cheese grating mills, you will need to cut the cheese small enough so that it actually fits in there, which is essentially adapting your code to work on the work on the GPU. It's an excellent metaphor because actually the block of Parmigiano in general, the most ever seen a guy like wheels, really huge wheels, they laid many, many kilos. There's no way that you can fit them in a small GPU in a small break. You need to break the Parmigiano in smaller pieces so that you can actually hold it and create it. And the same is with the GPU. And with the kind of speeds and type of amount of data that can take the points. All right, so this was just a little bit funny thing. I'm sure this metaphor, this joke can be proved in the years. But in general, there are some five take on messages from this panel. An introduction to Parallelization can speed up your tasks, but not any faster than the smallest serial pass. So if the smallest serial pass will always take eight minutes, you know, you're going to go faster than eight minutes. There's no way that you're going to make a fast class. And for example, in our case with the pasta, if you load data, what's commonly restricting you is actually loading the data from the hard drives. So this can't be speed up. You can boil the water faster potentially initially. That's more CPU power, but the pasta loading. That's eight minutes or in that case, in the example, cooking it is loading the data. That's eight minutes. Your hard disk is not faster. It can't provide faster. Exactly. So it's important then to think of your workflow. Sometimes you, if you feel that your processing is slow, and you need to see, okay, what is the kind of the smallest serial reading and processing. Because then if that is the, you know, the length of the process that maybe there's nothing that you can do that can, you know, take a bus to make a bus that will always take eight minutes. But maybe then there are things that can be run in parallel that don't need to depend on the whole process.  And when you can just stick with the previous execution of the code. The second thing I mentioned is that the benefit from Parallelization comes after modifying your code. So you really, you can just take off them very, very often. Most of the time you can just take your code and move it, you know, to a stove, to a kitchen that has multiple, multiple nodes. So, you, you need to not just modify the code and use this multi thread, multi processes. We also need to test that it's using this market. And this is also going to exceed the two and three that there are ways, for example, in the HPC cluster to check what was the performance of my code wasn't really using that for the four CPUs that I requested or 40 CPUs that I requested or was it in practice, I was only using one. And then the point number three is that sometimes we have special hardware that is built for Parallelization GPUs, but then once again, it will require us extra. And not all the libraries, not all the system automating the translator code so that it will use the GPU and work on the GPU. And sometimes it's not enough to just change the person. Sometimes it was something to change the full data loading, like, as I wrote there, sometimes we actually need to break the spaghetti that they can feed in this smaller memory. If your GPU only has a limited amount of memory, then yeah. Well, you need to divide your spaghetti into smaller pieces. That's how it is. So it's also the components, which is not the city compromising that I need to process this data only once we really need to spend all my time in rearranging the data reloading and it's a way of writing for the GPU. Or should I just wait, instead of doing it in one day, maybe I wait one week. And that's it. So this type of thinking on the resources that you need this needs to come from you. This to come this later to the specific. And then the fourth point is that requesting the resources from a big cluster, we should learn will inevitably require you. You are happy to request just one stove from one kitchen, and very often there is always a CPU that is available and you can start to you. You need to get to the conflict. But sometimes you really have a special cooking system that really requires 40 burners in the same kitchen for the CPU in the same mode for whatever reason, because you know the system software using. And then, you know, yes, there are those resources that are those notes available by you might need to be a little bit longer. And then once again, is it better to queue for two days and get something done in 10 minutes, or is it better to queue for one minute and get something done for us. Again, you know, this is not that there is no answer to this question, but it requires, you know, time and understanding what is the time and what is the cost. And finally, sometimes that's always remember that we might not even need organization. Not just because we need to do something that is serious, but also that sometimes the effort as I was just saying on rewriting everything into power doesn't really take forever still, it's beneficial to move the water on a remote HPC cluster. Because I see what was saying earlier, the need to keep my laptop running for 24 hours if it's a 24 hour processing job. When there's already a machine in the data center, there is already switched on with that there's already CPUs there already running, I will waiting for something to be so it's much more efficient for you for the environment. I don't think there is already on, there is already running where you can just send your strip there, most likely the data is also there so you don't need to start with the data around forgetting which version of the data is the most recent one. Let the machine, the remote machine to the company and then you can get the results to your local machine and make the pictures write the paper. And you can even work on on your computer without it being blocked by oh all the memory is used by my computation job or stuff.  So I think we're done with this. Some what funny introduction to Paralympic we're exactly in time. Let's check if there's anything on. I can make my presentation mode. Yeah, there was a. There were a couple of extra slides that I didn't include but you can look at them that some people might think well you know what I'm going to buy a mega pot. The mega pot is like okay now I have this mega pot I can make possible for people in just a single part. Yes, it's true, you can do that. I don't know how many times you have to make possible 100 people so this very expensive hardware is very useful if you rather than buying just for the set if you're sharing with others because today you need to make possible 100 people. And in general, I think this metaphor can be further. So maybe you people are listening to learn these things right now at the end of the three days you can think of other metaphors and metaphors to be able to understand and understand. Yeah. There's a question. How do you figure out what form of parallelism a code uses. I'm not sure if it's if it's analyzing code. Well you have to essentially look at it and if it's using some libraries. It's very likely that this is some kind of multi threading with within a certain framework. And inside some more low level library. If it's, if you did it, you or the top level of your code does it, then it's very likely that you can see it directly from the other code of the code itself what kind it's using. But if it's more the question what you should use and then this depends on the problem. There is no answer. Yes, like I would also add that like if you don't know if you're if you use already existing programs like like most of us do like who wants to write something that is available for everybody for free and made by somebody who's usually better at coding than you are. So personally I use everything. Like if it's if somebody has already created a tool that this does what I need it to do. I will happily use it because then I don't have to waste my time doing it and maybe do most likely doing it worse than the original author. So if I usually look at certain key buzzwords like they were already mentioned like multi threading multi processing MPI. Like if you see mentions about number of jobs or number of threads or number of workers it usually means usually means multi threading or multi processing and if you see MPI mentioned somewhere it means MPI. I usually like look at those buzzwords. If I see if I see GB you mentioned or CUDA it usually means that it's GB or it can utilize GPS. So just checking those can help you find out whether your program has a possibility of utilizing multi threading. And so many like let's say machine learning frameworks they often have like some sort of parallelism that you can use to do analysis just by adding not one parameter somewhere that makes it use multiple processes instead of one. And about what which framework is the best we are like Thomas said you you really cannot tell like this there's so many different problems and so many different ways of solving those problems that you need to pick the one for the for the situation. I didn't use the word embarrassingly parallel but it will come up in the future days but in general something that is embarrassing parallel it's like you really need to run the same process if I just change one. Because you want to test if you want to support the parameters like you want to test the script with the one with the two with the three. So that's where you should start if you know that you have a problem that could be splitted in this embarrassingly parallel workflow. Then that's the way that's where you should think. Yeah. So so in the also in the analog like if you can add more kitchens with more Italians that's like it can be it can feel stupid that you in hire more people to cook pasta instead of teaching one person to use multiple. Pasta pots but it produces the same results they or if it produces some results it's it's fine. So like adding more more cooks. If the and if the every cook does its own job. If it gets the same result it's it's fine like it doesn't need to be pretty if it works. Well I think the parallelism would essentially be there if you have a cook. If you if you can hire cooks on a temporary basis, then that works quite well. If you would actually have to hire them long term, but or buy new equipment, then it gets difficult. But since we can hire people for a certain tasks, yeah, should be fine. Okay, should we go to our break and then we can keep answering questions in the hack MD, even during the break. Sounds like a plan. Okay. Yeah, so keep asking and answering and we will see you back soon.


yo what's going on leafernation it's the leafer back again and welcome back to wager wheel episode number two and today it is a big one so you might be asking why is it a big one this is only the second episode like what's going to be so different so yes we will be doing the spin another wheel with the team and you guys know there is a loser wheel as well if i lose the game i have to spin the wheel to see what i have to do after the game not only will i be doing that but i am throwing and another thing on top of that that can really suck you're probably wondering what's that going to be and that is going to be me and this ultimate choice pack because you can see i have just enough to open up an ultimate choice pack and the thing is if i lose this game not only do i have to spin the wheel i have to discard whatever i pull in this ultimate choice pack now obviously if i do win i don't have to do the loser wheel and i get to open up this pack and keep anything that's in it so today is going to be a nerve wrecking episode we're hoping for a really good team from the wheel and we're hoping that we can get a win so i don't have to discard whatever i pull from here and as you can see we have a blank team it is time to start building here we go first spin of the episode we're hoping for a good team i do not want to have to discard this ultimate pack and the first line is a bronze line all right so um clearly clearly they want us to um to discard this pack okay let's spin for the second line and we hope the second line is obviously going to be better um my god we're already getting rammed here and we're okay silver player it is so i don't know man i guess we'll just go with any silver player at this point we'll go with a 73 probably our best bet there it's been in again here please give me some gold cards here another bronze line okay in all seriousness what's the odds we get back to back bronze lines my god this third line better be goaded uh third line please be better we'll stop stop being mean we get a choice my god thank goodness easy choice for me we're going with Austin Matthews European hockey event card finally a good card all right we're spinning for our well anyone on this line phone a friend okay hey man sorry to bother you for a video i need you to name any forward any forward any forward Jake Gensel Jake Gensel all right thank you all right mark says Jake Gensel let's go by Jake Gensel Gensel welcome to the team brother Jake Gensel over on the left wing all right we're looking for our right winger here no bronze players or silver players special card let's go special card right handed let's go with gila flurry easy choice okay the third line looking good safe to say they'll be playing a lot we're on to the fourth line here spinning the wheel my goodness please be good to us still we get another special card beautiful i don't care the wheel needs to be nice to us at this point looking on my team any forward this Jeremy Roenick is just screaming my name so welcome to the team buddy i know it's alone but it's a card it counts all right we got our left wing and right wing left here on this line give me another choice oh ea blessing me here all right shockaturi a yes please and last but not least the left winger will be a right winger on the fourth line we get a gold player we'll take it we will definitely take it any forward gold rating um i mean sydney crawlsby is screaming my name he's left handed though we need a right handed player uh let's go name mac then all right this is so our forwards uh shaped out first two lines absolutely horrible we are gonna get this demolish when these guys are on third line is way better we got uh la fleur matthews and gencil thanks thanks my buddy mark fourth line shockaturi a jeremy roenick and nape max so the last two lines definitely got saved they will be playing the brunt of the game that's for sure and now it's time to do our defense we're gonna get right to it spin it hopefully we get good defenseman man we get a pack line good thing we have enough for a 350 pack so whatever two defense when we pull from this pack will be a line for us here here we go all right some decent defenseman here please so far hickies are left defensemen tally segan that ain't bad my goodness did we pull another defenseman in this pack or no no are we having two left handed players on our d there it's looking like that's what we're doing here boys my god this first pairing looks absolutely horrible and second pairing here give me something good please silver player my god i have a really good feeling that we're gonna lose regardless looking for a right defenseman here gold player okay at least it's a gold player peter angelo looks like the best option here i guess so that's what we're gonna go with third pairing here give me a choice i got some great defensemen to use bronze player my god the wheel was just not nice to us today and last but not least the last right defenseman that we need it's a special card thank goodness gracious that special card is going to be none other than john carlson don't mind if i do our only good defenseman and now we're spinning for a goal or a goalie give me something good can't do a bronze line obviously so special card it is let's go uh the only special card goalie i have really is ben bishup so we're gonna go with him are we close to anything like dude we don't even have any synergies active we are gonna lose at least try and get spark active here if i'm a goalie with spark we're gonna throw them in waverith sure and we are gonna throw a spark coach in here if we do so at least we have one team synergy activated we have a spark coach beautiful so all in all this is a team we are gonna lose and we are gonna have to discard this pack i'll tell you that right now two lines of just bronze players uh the third and fourth line are gray with matthews laflare gensel katerie aronik and mckinnon defense looks horrible we got a silver uh bad bronze or gold player a silver petrangelo carlson in a bronze and thank god we got lucky in that but this is not going to go well this this this is going to be interesting to say the least we're starting off with the bronze line oh boy dry settle i bet you that's a team of the year two jr my god here we go boys right away just get the third line out here no no time to waste god these these cards are they feel so bad what a save bishop fourth line please third line did absolutely nothing oh my oh my god ben bishop thank you for being a goalie my god we're gonna have to play our bronze line here the second lines out here not good stuff third line's ready to go get them over the boards there we go good spam the poke i haven't edited any of these lines so this this is like an opportunity that i'm not gonna i'm not gonna get a chance with ah man you can't do anything with these bronze cards what a save ben bishop what a save oh i should aim that better here austin oh no i don't know if we block that if you save that my god my god my god these bronze cards are just so bad get that across there it is bang bronze player scores for us one one i am so frustrated you don't understand one one we're taking it i am beyond frustrated with this team i'm so quiet and so mad right now get that on the front oh jake ensal two one literally right after the first goal oh this feels so nice to score goals with this team i know it's my third line but still all right that gives us a little smile on the face after being super annoyed after one uh my god these stats dude we're up to one we're two for two on shots interesting let's go the flare one of the faster boys on the team gets right through get that award again so jake ensal the pickup from mark the phone a friend player scores the second of the game three two one so kateria get it over oh no why didn't ronix shoot that oh jr you gotta reach for that puck brother that's gonna be a penalty we are screwed we're giving up a goal here i think it's literally all bronze players on my pk bronze players yes sir we're screwed shoot that wow that is a you can tell that's a bronze player that straw was just horrible all right we killed it off with bronze players that's insane all right slow it down here man we need to just try to break in that is exactly what we're doing right now there you go ronik get loose of that oh i thought that was really gonna get through to kateria petrangelo we deflected it straight into babrowski's chest oh austin i don't know how we don't pick up that puck there we're they're calling another pen for hitting i love my hitting penalties in hl 21 end of two very even games still we i don't know man we just gotta hold on here we're up by two one more goal would obviously make me feel comfortable but uh let's kill off this penalty first and then we'll worry about the goals austin just killed the rest of the power play up there we love that man i'm gonna i gotta hold on to those pucks i just have to my team is not fast enough to react to stuff like that three two we're asking for it john carlson we need a big shot and john carlson finishes it back up by two what a shot john carlson we love that i'm sorry guys i'm not using the bronze players oh no i don't like that yeah i knew that was gonna happen too oh let's go we absolutely messed that up and we just got really lucky oh that got through that got through like three players no way game i gotta go back and watch that later i don't think that should have got through big defensive plays from the bronze defenseman jr off to the races where the hell did my car come from get that in front all the bronze player just missed we got 40 seconds here boys if we can get a whistle we're gonna need a huge time out that'll do it that will do it let's get the oh he quit we win dude why would he quit there's 30 seconds left he has a chance to tie the game up why alright boys might tell you a game was not fun this game was zero fun oh my god using bronze players is the worst thing ever just going to say that right now looking at the stats here i we played better we like that but now the best part we get to open up an ultimate choice pack and keep it so quickly looking at the stuff that is out 95 leondry settle 93 panarin 89 petry 88 ben master sets that are out on the team of the week that is out 91 sundeen we have a chance at all those guys and we get to keep whatever is in this pack ultimate choice pack 125 k pack 71 chance at 86 plus surely we get one out of this pack please be good to us you gave us the worst team ever and we won with it somehow so we're hoping for something good here Blake Wheeler was our best pull we're gonna go with the lower ended overalls here so we have a better chance at a higher overall that's just my thinking i don't know if it's true or not but second round all right primetime miles would alex martinez and a duggy hamilton let's go with those three why not third round we get to keep everything in here ea give me something good bernier haze skinner and patrice bergen sorry patrice we we are not gonna take you buddy we won an 86 plus my friend fourth round here we have daemon severson tyler sagan quinn hughes antifoli give me hughes give me those three last round give me a purple yei bag man 86 plus 71 chance at it surely kutchuck has his master set out now feels bad man oh my god not anything good we get to keep these i guess but like nothing good anyways guys that is it for today's wager wheel episode if you guys did enjoy be sure to drop a like on it for you guys a new round here make sure to hit that subscribe button we're on the road to 4 000 subscribers um this wager wheel episode was so interesting i hated the team we ended up winning we got our ultimate choice pack and pulled nothing so we might as well just lost that game looking at it now i hate that anyways guys thank you very much for watching and i'll talk to you guys later peace


The opening ceremony we held today on Thursday the 12th of June was to open the Pacific Partnership 2014 here in Timor-Leste, a small part of the grand plan of Pacific Partnership throughout Southeast Asia. The history of the nation are largely Portuguese as a colony, independence in 1999 after 25 years of occupation.  United Nations mission ended about two and a half three years ago so they're now at the point where they're finding their feet they're starting to develop capacity building in terms of police, defence, infrastructure, health, education and so on. The Defence Cooperation Programme is responsible for a number of functions we train in specialist areas such as engineering, basic medical, logistics, communications, maritime and English language program and we also advise the Department of Defence in strategy and also governance. The importance of helping them develop a course is they're a very close neighbour to Australia. They were pretty much an ally so to speak during World War II when we're involved here in operations against the Japanese without their help the mission would have been a failure. So to a degree we have a debt of honour to Timor-Leste and of course there is that goodwill that extends to your neighbours. The Pacific Partnership is truly you know it's a great exercise it brings a lot of goodwill. Soldiers, sailors, airmen from America, from New Zealand, Australia, other nations actually get to work in with the local population like in this engineering site here in a medical facility side by side they get to learn a bit of a local language and that's something you wouldn't get on a normal exercise back home.


What's up everybody, JC here for jasperjasecasebrakes.com. 2020 Top's Inception Baseball just sold out, one box break, number five. And again, this is a break to win two spots in transcendent. First, Randomizer's customer names and teams and we're no Cardinals, so we sold 29 spots. At the end of the break though, we get a new dice roll in Randomizer. Randomizer customer names. And top two, get a spot each in the transcendent Hall of Fame Baseball, which is a $450 spot each. So again, from one of the last breaks, we're gonna use a psychic sign, so we're gonna use the physical dice to randomize customer names and teams. So here you go. One or three and a one to make four. Now let's switch scenes. So here you go, customer names, four times. One, two, three, four. Here you go, one, down to 29. Now let's go four times here. Four times, one, two, three, four. Royals, down to the Brewers. All right, Royals at the top there, Brandon with the Rangers, Tim with the Padres. You got Rockies there. At left, Aletta, I think it is, sorry. Brandon with the Reds. Aletta with the, or Attila, I'm sorry. It's blanking in my mind right now. Phillies, you got Blue Jays there. Cubs, Marlins, Giants, Indians, Rays, White Sox, Metz, Jarrett with the Angels, Brandon with the Dodgers, Joe with the Tigers, Jacob Yankees, Tim, A's, Rob with the Mariners, Quinn, Brandon, Orioles, Nationals, Pirates. I have brain farts sometimes. Leo, Joe, Braves, Astros, Twins, Brandon with the Brewers. Trade window if you guys want any trades. I don't have anybody who wants a trade. Let's go to the trade window. Attila, Attila, Attila, yeah. All right, here we go. All right, we got our Rosarena, which I'll randomize again because Cardinals did pop out. So I don't go to number three on the randomizer. Grisham, Jake Rogers, Toro, Soto, Amonteros, and Danny Giants in the 245 for the Toronto Blue Jays, which Blue Jays is Jacob. Very nice. And you got a handful of rookies in here. Only Cardinals, not a rookies on Soto. All right, so top two. I'll get this spot in transcendent number three. We'll get the same little Cardinals hit right there. That's two in a row with the Cardinals. All right, let's switch scenes. Young customer names. Number top two, get the transcendence spot. Number three gets the card. Whoa, okay. One, three, four times. Good luck, four times. One, two, three, four. Boom. Wow, there you go. Brandon, Brandon, top two. And there you go. Number three is going there for the Cardinals. Card, appreciate it. Jaspy's casebreaks.com. Another exception, number six I believe is up. Winston Spots there in transcendent.


Giving a traditional style academic speech from home can be a real challenge. There are several environmental and technological barriers that need to be overcome. That said, there are a few things that you can do to stoke out your space and set yourself up for success. In this short video, I'll cover how to set up a home speech space. So today, we'll discuss how to select a space, how to fine-tune your lighting and sound, and ensure your technology is set up to record or stream a good take. Let's get to it. Let's talk about selecting a space to give your speech in. You can reduce many of the challenges by picking a space that has less issues to begin with. When choosing your speech location, you need to factor in two key things, size and environment. Let's quickly discuss both. First, when it comes to room space, you need to make sure that you have enough floor space to move around in. The delivery of a basic speech often calls for a speaker to move to different physical spots on the stage as they progress through the speech. You should aim to find a space that is at least eight feet long and three feet deep. Most bedrooms and offices meet these requirements. Slightly larger spaces will give you more freedom of movement and can make it easier when it comes to combining transition statements and stage movement. Living rooms are the most popular choice here, though I've perhaps seen folks make less conventional options work as well. In addition to having adequate room, you should also consider the overall environment. If possible, aim for a space with a neutral background that you can speak in front of. Blank walls are fine, but it can also work to speak in front of more unique backgrounds. Just make sure that there's nothing too distracting or inappropriate behind you. Now that we've discussed picking a space, let's consider some of the ways that you can improve your lighting and sound. First, whenever possible, try to choose a space that has both good front-facing lighting and decent acoustics. During daytime hours, facing an open window is an easy way to get naturalistic light for your video. If that's not an option, you can make artificial light sources work well too. At the most simplistic level, you can offset a single light source, like a lamp, without a shade to light your speech. This setup is called one-point lighting and should work for most situations. Just make sure that the bulb you choose is bright enough to light the room. If you're looking to take your lighting to the next level, you can opt for a three-point lighting setup. Like the name suggests, a three-point lighting setup relies on three lights pointed at the person giving the speech. The first light, known as the key light, should shine directly upon the speaker from a slightly elevated position. This light will serve as your principal illuminator. You want this to be the brightest light of the three. The second light is called the fill light. This one should also shine right on the speaker, but from a side angle relative to the key light. If possible, situate the fill light at a lower position than the key light. Eye level works well. The fill light helps by balancing the key light by illuminating shaded surfaces. This light does not need to be as bright as the first one. The final light is called the back light. Out of necessity, you'll want to set this one up off-camera, lighting the back of the speaker. This helps provide contrast between the speaker and their background. Combined, the three lights should give you an effect not dissimilar from what you would see in a professional video or photography studio. Regardless of the lighting setup you opt for, make sure to avoid standing with a bright light directly behind you. Otherwise, you will be very difficult to see. Once you have your lighting dialed in, you should turn your focus to acoustics. First, make sure to eliminate any obvious sources of noise. Silence phones, turn off TVs, and make sure your smoke alarms have fresh batteries and are not chirping every 60 seconds. If your space is very empty, you may find things can get pretty echoey. If possible, aim for a room with some furniture or other items. If that's not possible, you may want to consider using a body microphone to improve audio pickup. While it can be difficult to control, you also want to try your best to limit external noise too.  Now that we've covered the basics of setting up your space, let's discuss a few important considerations for setting up your recording technology. Let's begin by setting the frame. You want to position your recording device so that it does two important things. First, make sure that you are visible from at least the hip up. This is important as your gestures and movements need to be able to show up on screen. Second, you want to make sure that the top of the frame leaves a little bit of headroom, but not so much that you can see the ceiling. Aim for a little room above your head, but avoid going above the top of the wall behind you. You should also try to keep distracting elements like hanging lights and ceiling fans out of the shot. Third, when setting up your recording device, you want to make sure that the camera is far enough away that you have room to move about the frame between main points. This means that your 8 plus feet of stage space should be visible from your fixed camera location. This lets you move between main points without the need to have a camera person panning the camera to follow you about the speech space. Next, let's look at your camera setup. First, it is important to make sure that your device is set to record in landscape mode. Most computers, camcorders, and digital cameras do this by default. However, if you're using a phone to record your speech, you'll need to orient your phone horizontally to ensure that the full frame is recorded. You may also want to pick up a simple phone tripod to help make this process a little easier. With that in mind, you should also consider the height of the camera. If it is placed too low, it can give the appearance that the speaker is towering over the audience. Aim to place the camera as close to your standing eye level as possible. You may need to place the camera on top of some stacked boxes or use a tripod to get this right. This will help you make good virtual eye contact where the camera lens can be treated like a face in the audience. It also avoids the effect that you are looking down on your virtual audience. During your speech, try to connect with your camera lens as much as possible, or at least as much as you do any other of your virtual audience members. Next, let's talk about your microphone. Most microphones included on computers and phones work well enough to record a speech, but there can be some problems. If you use an on-device microphone, you should make sure that it's not covered up and test it beforehand to ensure that you can be heard. Some laptops, most notably MacBooks, situate the microphone near the keyboard. If this is the case with your computer, placing anything on it during recording can make it very difficult to hear the speaker. Some built-in microphones can also have trouble picking up the speaker if they are not within a couple feet of the device. If your stand is causing you trouble, consider an external microphone or even using a wireless mic. Wireless headphones like AirPods are also an easy way to accomplish this and can improve audio quality. Fortunately, there are a plethora of affordable wireless headphones and external microphones available these days. Once you have your setup dialed in, you should make a short recording to test it out. This will let you diagnose any issues you might have and help you avoid recording a take of your speech that just doesn't work. Well, folks, that wraps up some key considerations for setting up an ideal speech space. Giving a speech-style presentation online can be challenging for a variety of reasons. Dialing in the perfect speech space can go a long way to reduce some of the stress. Thanks for watching.


Good morning, we will discuss now functions of random variables or transformations of random variables. So, this is a topic that is motivated as follows. So, you may have a random variable which maps the sample space to the real line. However, you may be interested not the exact value of the random variable itself, but some function of it. So, you may measure x directly, but you may be interested in the distribution and statistical properties of let us say x square or log x or something like that, because your application may demand that log x is of more importance to you than x itself for example, that very often happens in applications in engineering and statistics. So, this is the motivation for studying functions of random variables. See you should recall that random variables are themselves functions. So, when we talk of functions of random variables, we are talking about composition of two functions.  Now, you may be interested in some function of x. So, let us say that f is some function that takes values from r to r to begin with. So, what f does is it maps. So, this is another real line and this guy maps. So, this is x of omega and f will simply map. So, this is f. So, once omega realises x of omega is a fixed number and f will map that further to another point here, which is f of x of omega. Now, if you look at the overall mapping. So, if you do not look at this real line at all, this overall map will be f composition x. It looks as though you are mapping it twice. So, essentially when you are talking about function of a random variable, you are talking about composition of two functions from omega to r. So, the problem is. So, if you are given the probability law or the CDF of x, I want to determine the probability law or the CDF of f of x, f of capital x. Now, before that we must address whether this composed mapping f composition x, whether that is a random variable at all. So, if you want to speak about the distribution of x, we know how to do that. But, if you want to talk about the probability or CDF of f of x, we have to first establish that f of x is a random variable. What does that mean? It means that if I take Borel sets here, its pre image must be f measurable here. Only then this f of x a random variable by definition. If I just look at it as one mapping f composition x, under that composed mapping I must have pre images of Borel sets are image. Now, one see one sufficient condition for that to happen is that if f were to be a Borel measurable function, which means that if I have Borel sets here, its pre images here are Borel sets. Suppose that is the case, then the pre image of that will be an f measurable set.  Is that clear to everyone? So, what I am saying is if f is Borel measurable, which means I e pre images of Borel sets are Borel sets. Then f composition x is a random variable on omega f p, agreed with that. What I am saying is f is a Borel measurable function or simply a Borel function. It is that is like saying that if you take any Borel set here, its pre image under f must be a Borel set on r. And of course, if you have a Borel set here, its pre image must be f measurable, because x is a random variable. So, is this statement with everyone? So, the problem is more precisely stated problem is if you have a Borel function f, f x is a random variable and I want to determine the probability law of f of x given the probability law of x. That is the problem confronting us. And now I mean this is you can even speak about now functions mapping r n through r m in general. You may for example, have n random variables x 1 through x n defined on omega f p and you may want to talk about f of x 1 x 2 x n.  Then you have sample space then you have some r n here. So, you have some x 1 x 2 dot x n. So, you have a map from omega f p the same underline probability space is being mapped to some vector in r n.  And now you have some function this could also be r m let me just draw f. So, this will be f of dot dot dot this will be f of x 1 x 2 x n of omega. So, here to you want to take any Borel set it is pre image must be a Borel set in r n. And of course, every Borel set in r n pre images event is an event.  Again here the question will be given the joint law of x 1 through x n what is the CDF of f of x 1 x 2 x n. So, that is that is what we are going to discuss. So, are there any questions on the setup everything so far. So, our agenda will be as follows first we will consider some special case of special cases of these functions. So, first you will consider maximum and minimum maximum and minimum of random variables which is actually fairly easy. Then we will consider sums of random variables because they are the maximum and some are like of great importance. And then we will consider general transformations some f some generic f. So, our agenda will be first we will discuss maximum and minimum then we will discuss sums of random variables. And finally, we will address more general transformations or more general functions.  Let y n is equal to min of z n is equal to max of. So, you are considering n random variables. So, a finite set of random variables x 1 through x n whose joint CDF is given to you. And you are looking at the min of this x 2 x 1 through x n and max of x 1 through x n. So, what does this mean for every omega. So, whenever omega realizes I have x 1 omega x 2 omega x n omega realizing as real numbers. For each such omega y n of omega will be the smallest of those numbers and z n of omega will be the largest of those numbers. So, for some particular omega x 3 may be the smallest, but for some other omega x 1 may be the smallest and similarly for z n. So, you are looking at the distribution of the smallest of these x i's and the largest of these x i's. So, now are these even random variables we have to see this is some function this is like that f of x 1 x 2 x n max is max of x 1 x 2 x n a measurable function or for that matter is min of x 1 x 2 x n a measurable function that is the question. Once we say yes it is a random variable then we can go ahead and talk about its CDF or probability law. So, we have to first establish that note that y n and z n are indeed random variables since. So, what do I have to show I have to show that pre images of Borel sets here but, here actually because this is such a simple function there is a easier way of doing this. So, for example, if I take let us say y n. So, the event if I take the event that y n of omega is less than or equal to little y. Well maybe I should do it for z n first little bit easier with the maximum let me do it with z n little z n little z. So, I want to show that that is z n is a random variable. So, if I show that this is an f measurable set I am done because this means that the pre image of the semi infinite interval here is impact an event any questions. So, if I show this is a valid event I am done, but z n is the. So, this is the event that this is the event that maximum of x 1 omega dot dot x n omega is less than or equal to little z. Next note that the maximum is less than or equal to little z is equal into saying that each 1 of them is less than or equal to z that is the key step. So, this is equal to the set of all omega's for which x 1 of omega is less than or equal to z and x 2 of omega less than or equal to z dot dot dot x n of omega less than or equal to z. So, when I write comma again I mean intersection what actually mean is intersection i equal to 1 to n omega such that x 1 of omega less than or equal to z and so on. So, what kind of a set is that. So, this is some semi infinite cuboid in r n and x 1 to x n are random variables and in r n if you take any semi infinite cuboid in r n it is pre image has to be f measurable y x 1 through x n are random variables correct. So, if x 1 through x n are random variables pre image of borel sets are f measurable in particular the semi infinite recta cuboids on r n are the generating class.  So, these are this is an f measurable set for sure right because x 1 through x n are random variables. So, this is an f measurable set correct are there any questions. So, max was very easy that is true, but every time. So, what you are saying is that z n is always one of the x i's, but it is the i will be different from different realization. So, what I am saying is that. So, if for some omega. So, for some particular omega it may be x 3, but for some other omega it may be x 5 right. So, the index of the maximum itself is a something random correct you cannot fix that index it is not as though a particular x i is always the maximum right correct. So, it is now you cannot claim that z n is always one of the x i's and x i is a random variable. So, z is a random variable that is not correct because it is true that z is always one of the x i's, but that i itself is random it becomes depends on omega correct. Yes, but it is enough to prove for the generating class they have pre images of semi infinite intervals are f measurable. Then you can show that pre measures of all Borel sets are f measurable. Now, similarly how will you show that y n is a random variable let us do this here how do you show f. So, y n.  If this is an f measurable set then necessarily y n of omega less than or equal to y will be a measurable set because it is only the compliment right. And why am I taking greater here because I am dealing with the minimum right eventually I am going to write. So, I am going to write omega this is omega such that x 1 of omega greater than. So, well first I should write minimum x 1 omega dot x n omega greater than y right. And if you are saying that the minimum of all these guys is itself greater than y then each of them must be greater than y right the equivalent right. This is equivalent to saying that omega this is omega such that x 1 of omega greater than y comma x 2 of omega greater than y that dot x n of omega greater than y all right. So, again commas mean intersection. So, is this step clear this is the key step now. So, this is now this is an f measurable set right you are just this you can show us an f measurable set because this is like. So, this is like a semi infinite the cuboid, but it is pointing the other way right you can show this is a f measurable set. So, this is an event all right this is f measurable and again this is f measurable. So, we have proved that both y n and z n are valid random variables right. So, now our task is to determine the c d f of this z n y all right. So, let us. So, actually these expressions themselves provide a very easy way to calculate the c d f in fact. So, I know this is the event whose probability I am after all right my c d f is after all the p of this event correct. So, next f z n of z will be equal to probability that omega for which z n of omega less than or equal to little z that is equal to probability of omega for which x 1 of omega less than or equal to z dot dot dot x n of omega less than or equal to z right. So, I am just putting p here and p here right. Now, what is this do I know this I want this right do I know this this is the joint c d f evaluated at z z z z z right.  So, you are given the joint c d f of x 1 through x n if you replace each of the arguments with z you will get the c d f of your maximum that clear. And similarly, I want to talk about the c d f of y what would I do I will do 1 minus this right or rather if I just take the probability of this event I will get 1 minus the c d f correct with the complementary c d f as it is called right. And so, I say f y n bar of y right this is complementary c d f right that is equal to 1 minus f n of y f y n of y rather that is equal to probability omega for which x 1 greater than y dot dot dot x n greater than y. Now, what is that it is not really 1 minus this is the right this is I mean this is really that. So, this is like the complementary joint c d f right which you can calculate from the joint c d f right. So, in the max you directly get c d f in terms of c d f here you get complementary c d f in terms of joint complementary c d f correct. So, maybe I should just write this as f bar x 1 through x n of y y y this is the joint complementary c d f. So, if you give me the c d f I can find the c d f of the maximum and minimum right. So, this is this assumes nothing further right this does not assume that x i's are when x i's can be any random variables they can be some of them could be discrete some of them you know it anything is possible you are just given some joint c d f right. I am not assuming anything further about what the whether they are discrete or continuous or otherwise or I am not even assuming anything about the joint distribution I am not assuming independence in particular right this is some arbitrary joint distribution. So, this is a completely general expression. However, if you consider the special case where x i's are independent x 1 x 2 x n are independent then these formula these all this formula will simplify right. So, in particular for example, for the maximum what would you get you have the joint c d f evaluated at z z z, but the joint c d f will factorize product out correct. So, this will become f x of z power n right. So, here again you will have the complimentary c d f to the nth power right in particular if x 1 x 2 dot x n are independent. So, first of all first let me do the case when they are independent. So, I will have f z n of z will be equal to product f x i of z z is equal to i is equal to 1 through n that correct. Because my joint c d f will factorize right further if x 1 to x n are independent and identically distributed i i d means this is the first time we are encountering this I think independent and identically distributed. It is just as the word says right if they are independent and they have the same c d f's which means all this f x is of z are the same function f x right with c d f f x then the maximum will be distributed as this will be simply the. So, all these are the same function now. So, now it will be nth power right. So, if they are independent you will multiply the c d f's if they are independent and identically distributed you will take nth power of the common c d f well I should say z here right. Similarly, you can do the same thing for the minimum. So, in this case what would you get you will get you will get f. So, the complementary c d f of y as the product of the complementary c d f of each of these x's and in the case when they are identically distributed you will have the nth power of the complementary c d f right. Next we have for the minimum we have f y n of y is the bar is equal to product. So, I am still with the independent case product i equals 1 through n it will be f x bar of f l f x i bar of y correct. So, because of this formula I had here and this f y f y f bar will product out the complementary c d f also product out right when x 1 to x n are independent f y n bar of y is equal to f x bar of y to the nth when x 1 to x n are independent and identically distribute i i d. So, far. So, let us consider one or two examples. So, that completes the theory bit. So, far any questions on this? . .Yes, that is what we need to prove, but here there was an easier way out. So, you directly prove that pre measures of semi infinite intervals are events you exploited the fact that is x 1 x n x 1 through x n are random variables. Yes, it is it is it is. So, no it is a random variable with respect to I mean there is random variable just means the measurable function finally, it should be a measurable function of omega right. You can prove it with here where you want whether you want to prove that f s borl from r to r that is fine or if you want to prove that pre measures of all borl sets are f measurable that is also fine. Here we proved that pre measures of semi infinite measure events are semi infinite intervals are f measurable that is enough. See usually definitions of measurable function we make in terms of pre measures of borl sets, but it is very difficult to verify that right, because borl sets can be so complicated and their pre measures can also be quite complicated and you have to do this for all borl sets. So, what you do is just verify for the generating class and it then follows that it holds for all borl sets right it is something you need to prove examples. Let x 1 and x 2 be independent identically distributed let us say uniformly distributed random variables. So, you remember uniformly distributed random variables right in 0 1 let us say let z is equal to max x 1 x 2 and y is equal to min x 1 x 2. So, you have x 1 and x 2 inducing a uniform measure on 0 1 and they are independent. So, they are independent and identically distributed random variables and you are looking at the maximum of the 2 and minimum of the 2. The question is how are they distributed. So, are there any guesses on how qualitatively they may look. So, you are taking 2 identically distributed random variables on 0 1 independent and you are taking the larger of the 2 and the smaller of the 2 sorry qualitatively. See it is clear that both z and y have to take only values in 0 1 right. We are talking about 2 values 2 random variables which are both taking values in 0 1 right. We know that z and y have to be dependent because that is bigger than y right fine. And if you look at the distributions of distribution of z what do you think it will look like it will be. So, it will be more towards 0 or more towards 1. So, z will have more mass towards 1 and y will have more mass towards 0 that what that is what you would expect right. So, let us actually verify this with our formula that we just derived here. So, we have f so what is f x.  So, f x 1 of x will be x is not it. So, will be 0 x and 1 depending on whether x is less than 0 or x is in 0 1 or x is greater than 1 right this is our CDF. So, from our formula there you can immediately see that f z of x f z of z if you like f z of little z will be equal to f x of little z to the square right f x well by f x I mean either f x 1 or f x 2 they both the same right. So, this will be equal to 0 z square or 1 depending on whether z is less than 0 z is in 0 1 or z is bigger than 1 correct. So, that looks like if I plot capital F z of z will look like a parabola right and you can see clearly that this function is differentiable except these two points right and you can actually get a density as well right after all x 1 and x 2 are continuous random variables and z also turns out to be a continuous random variable. So, your density pdf f z of little z will be what be 2 z is not it 2 z if z is in 0 1 0 otherwise right. So, your little f z will be like this little f z will be 0 outside this and here and here. So, at 0 and 1 there is no derivative, but you can define it as whatever you want does not matter and in between it is equal to 2 z right. So, it will look like that right is equal to 2 right. So, the pdf looks like and that is a valid pdf obviously. So, you see that. So, more of the mass is towards 1 right it is likely to be closer to 1 the maximum is likely to be closer to 1 and similarly you can prove that f y y is the minimum right. So, f y of little y will be equal to. So, f y bar of little y will be equal to 1 minus y square is not it will be 0 sorry will be 1 1 minus y whole square or 0 depending on whether y is less than 0 y is in 0 1 or y greater than 1 and from here you can get the density again is this correct or missing something. So, I am just taking the complementary cdf of x which is yeah which is 1 minus y and I am squaring that I think this is correct right. So, the pdf will be now the if you plot the pdf it will be the derivative of this that will look like look like that right be twice 1 minus y it will look like that. So, the minimum is has more mass towards 0 as you would expect is that clear any questions yes. So, this is complementary cdf. So, 1 minus that will be your cdf right and you differentiate that you get a pdf ok. Let me just do 1 more example in the 5 minutes that remain. So, example number 2 let x 1 x 2 dot x n be independent random variables which are exponentially distributed with parameters. So, exponential the parameters by 1 positive parameter lambda right. So, I am going to give you the parameters lambda 1 lambda 2 dot dot dot lambda n greater than 0. So, I am telling you that these are exponential random variables they are independent, but they are not identically distributed they have different parameters. So, in this case what say f. So, if you have f x i of x will be equal to 1 minus e power minus lambda i x for x greater than or equal to 0 correct. Remember this cdf exponential right. So, this is the cdf right and for x less than 0 this is 0 it looks like that. Now, if you look at y is equal to minimum of x 1 dot dot x n I am looking at the minimum of this independent exponentials. So, now what happens. So, I will have f y of y will be equal to product well I should say f y bar is not it f y bar is equal to product i equals 1 through n f x i bar of y correct because now x i's are independent they are not identically distributed. So, I cannot take nth power correct. So, this will be equal to what say f x i bar will be e power minus lambda i x. So, I am multiplying. So, I am multiplying product i equals 1 through n e power minus lambda i y correct. So, now and this will be equal to what e power minus sum over i equals 1 through n lambda i times y. Now, what does this mean it means that y is also an exponential right y is an exponential with parameter sum of the lambda i's. So, if you take. So, what does it mean. So, if you take n independent exponentials with parameter lambda 1 lambda 2 lambda n then the minimum is distributed as also an exponential with parameter sum of the lambda's right. So, y is I can write exponential with parameter sum over i lambda i's you can also compute the distribution of the maximum. What would you do you will just multiply now the complimentary c d f the c d f themselves you will multiply all these guys. But, that you will not get anything nice you will get some distribution you will get the product of all these guys right. So, the maximum will be some distribution the minimum is an exponential. So, the minimum of independent exponentials is always an exponential and the parameter is the sum of the parameters of the individual ones. So, this has a very nice practical interpretations I remember that I mentioned exponential is this memory less distribution that arises frequently in let us say radioactive decay right. So, each let us say that each of these corresponds to the i th sample they may be there all different radioactive elements or something. So, they emit alpha particles at different rates right. So, and the each sample i th sample let us say has inter emission duration is exponential with parameter lambda i. And you let us say that they all are emitting at one point at in some place and the minimum of these is the time to the first emission correct. You have all these independent samples emitting. So, the if they are all in independent exponential times the time duration for your first the time for you to see the first emission coming out is also an exponential right that understood. So, yeah. So, this has lots of nice properties right. So, this is this these inter emission times being independent exponentials is a very well known process called the Poisson process. So, it has lots of nice properties we will not study it in this course in any great detail, but there is another course called stochastic modeling right which will be offered next semester that is half the course is about these Poisson processes. So, that is just f y i, but all you need to know now is that the minimum of exponential is also an exponential independent exponential is also an exponential ok I will stop here.


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