Massachusetts to, I want to say, Cincinnati? Or did I? I don't remember. But I had to go to another state for, what was the thing, geek squad induction?  Cincinnati, Ohio, or, no, no, no, no, no. I think it was Connecticut.  Yeah, I had to do that and I took the Amtrak and, I mean, it was okay, you know, ride on the Amtrak. It was pretty chill. They had Wi-Fi. I was playing Binding of Isaac, my computer. Very bass game. Yeah, it was a pretty fun game. I just, I didn't keep up with it, though. There's been too many, like, I feel like after Rebirth, I don't know how to play that game anymore. I don't know any of the metas and, like, I don't know what stuff does. Yeah. They only had, um, they had two more updates of that, maybe after Birth and Repentance, and then they stopped. Yeah. Well, I guess that makes sense, because, I mean, it was like, it seems like they were running out of, like, bosses to fight, because in Rebirth, I think Mega-Satan was the new boss they had.  It's like, who else is left? I think there's, like, the Beast as one of the new ones. Yeah, now you fight, now you fight the Angel and you can fight, um, Blue Baby and you can, which is basically Isaac. Uh, I think Blue Baby was, well, hang on, no, Blue Baby was in older games. They added, what was the other one, Hush. I forget which one Hush was. Yeah, they added the Hush, and then there's, like, um, I think it's Dogma, right? Dogma is the TV that, like, transforms into, uh, the Dogma where... The horseman in Rebirth or the afterbirth? That was before Rebirth, I think. It was? Yeah, yeah, that's like a mini-boss. Yeah. But, you know, the real question with this high-speed rail is how is our stuff going to stack up to the rest of the world, right? I think China has the most high-speed rails. I'll just read what this says. As of 2021, China had more than 155,000 kilometers or 96,313 miles of railways, the second longest network in the world, okay? Oh, so yeah, they have the most high-speed rails. By the end of 2022, China had more than 42,000 kilometers of high-speed rails, the longest HSR network in the world. And I'm pretty sure they did this in, like, two decades, because I think pride might be listed in here. Like, prior to 2006, they didn't really have high-speed rails. Yeah, this goes all the way back to, like, the United States. Okay, taken up by the ministry. Or inspection. I don't know, but like, I think it was prior to 2006, they didn't really have that many high-speed rails, or they were like 10th in the world, and then they'd be number one. Yeah, because in a socialist country, it's very easy to divert resources to public projects. True, yeah.  It's very simple to do. We caught some people googling freedom, and now you build railway. Yeah, I think that's going to be the big issue that we run into here in America, because there's so much private land ownership that you got to buy those people out to build. Yeah. Trains. Or go around them. Yeah, too. The biggest problem, though, is that the government tries to run, like, Amtrak as a business to make it profitable. And, like, generally, that's a good idea, but if you're running a public service, this is the same issue with the T, is they try to run the T like a business and make it profitable. It's like, motherfucker, it's a public service. It is not meant to be profitable. It's meant to provide a service to the public. Your revenue comes from taxes. Right. So that's a problem. And generally, like, you know, people like Alan Fisher on YouTube, their easiest solution is just do everything in house, right? Just hire people directly and do everything in house. And, like, that's a generally good idea. But you underestimate the expediency of politicians. The reason that they usually hire contractors and the reason government contractors exist is because the government doesn't want to take, you know, five to eight years to actually develop a workforce that is skilled in making these rails and maintaining the trains, because that's way too long, right? The election cycle is way too long for that shit, right? They don't care. What they want is they want to lay down track as quickly as possible so they can show off to their constituency that they've actually done something. The easiest way to do that is to get a contractor who already has the skill the labor and the materials needed to do that.  Oh, yeah.   Oh, yeah, man.  I'm trying not to commit treason here, but, like, I used to do some budgeting back when I was, like, the only IT guy at my, like, a financial firm. In my budgets were, like, typically, you know, like, tens of thousands at most. Budgets here easily, like, a thousand times. Like, we're talking tens of millions of dollars.  But, you know, that's the thing. Like, America, like, because it's tax money, right? Like, public money is tax money.  Like, spending it in Ukraine, spending it in Israel, spending it, and not even, like, now in Israel. Like, for years now, we've been given billions of dollars to Israel. You know, we give billions of dollars to so many different countries just to be police of the world or just to, like, maintain, I guess, our, what is it, our strategic bases and stuff where we can basically take on the entire world in a military conflict all at once and still win. It's like, man, it'd be pretty cool if we could just, for a little while, for a few years, make a deal with China, make a deal with Russia, make a deal with, you know, all of our enemies or whatever. Be like, hey guys, so we feel pretty bad about the fact that our internet is like the 20th fastest in the world and our railways suck. Just give us a few years to catch up to South Korea. Let's make high-speed internet, high-speed rails, put high-speed internet on the high-speed rails. And then we're going to go back to being the police of the world.  There's no way. I think China's too posty to annex Taiwan. Yeah, because we exist. Because we exist. You know for a fact that Taiwan has bombs underneath TSMC of China invades. But who's bombs are they? Well, they're bombs they bought from America. Right, and so does it need, because I feel like America doesn't just give you bombs. I feel like America's like, okay, here's a bomb, but someone here in a bunker somewhere has to press a button to arm it. Otherwise, it's just a metal tube, you know? I don't know, I don't know how any of that stuff works. But I hope we get some heckin' high-speed trains. I was doing a little bit of research on this to try to, you know, talk about it since people are interested. Oh, it won't show me the information now. Really?   Because I was looking at the same page yesterday and I was able to get the stats. But I wonder, it's because you went there too much. I wonder if you can pull it. Let's see if you can pull it and share your screen. All right, let me try. I would surprise some of the countries on here. It's like Italy has better trains than us. Doesn't California, by itself, have a bigger economy than Italy? Well, yeah, because you pulled that up. Because European countries aren't as... The problem with America, and part of it is lobbying by oil companies, because they're fuckin' huge in America, is that America is a very car-centric country. You know, I know, because we both have cars. Everything here is designed and has been designed for the past 60 years to be car-centric. Countries in Europe are both... The countryside's are typically... They've been populated for centuries.  Yeah, that too.  In some cases, three times. They rebuilt everything in the 40s. Yeah, in some cases, multiple times if you're in the Balkans. Their investment into rail was essentially... Because a lot of European cities are very congested, because they've been... They were designed as walkable, because back in the day, you had two methods of getting places.  And it generally makes it really easy, because you just make a high-speed rail between Rome and Florence. Like, where are you going to need to go in between Rome and Florence? Maybe a couple of different stops, but generally only people who live in those areas. Because everybody tends to live around an urban area. Right. Like in Iceland, for example.  And Reykjavik is probably the size, I would say, of like Portland, Maine, or like maybe Cambridge, Massachusetts. Okay. In terms of like how big the actual city is. Right. So in America, it's like from our founding history, like people would move out to the countryside and they would live on homesteads away from the city. And the only way to get there were via horse and buggy, and eventually a car. And because that's how people have lived in America, things are so spread out that this is my problem with high speed rail people investing into it, is their scope is very limited. And it's only useful for places like California, where you have like a string of metro areas that are highly populated, like San Francisco, San Diego, Los Angeles, Oakland, essentially San Francisco. Right. All up that West Coast. And because you can, people usually travel between those all the time, right? People travel between San Francisco, Los Angeles, all the time for business. Same on the East Coast where you have like, what is it, boss wash where it's like Boston, New York City, Washington, Baltimore, right, that whole Eastern keyboard. That works.  Middle America. Like how are you going to get there by a train? You really can't. Yeah, it's not going to happen. It's going to be, it's going to be pickup trucks. Yeah. Trucks and well, not necessarily all pickup. But it would reduce the traffic between those metro areas, which I think does have benefit. Oh yeah. Cause I mean like out, well here in Southern Virginia, it's like there generally is not that much traffic. Like there's usually not, you know, gridlock traffic or like usually the only time there'll have to be some traffic is if there's a log truck that's driving through town. So, you know, really, really big and they tend to go a little bit slower. Also, he isn't using CREX. Not going to make it. Abram. All right. I have, I have the page up. Sweet. Okay. Share a screen. I think it'll still use. Camera. Hopefully it doesn't. I have to stop the screen. I think you have to stop sharing, sharing your screen. I think. See what we got. I just want to look over that because like Italy, has the fourth or fifth and as far as GDP goes. So, California's GDP for 2022 is 3.59 trillion. Italy's is 2.1 trillion. So it's almost double the GDP of Italy. And we know how, I mean, I don't know, maybe Italians are better at building trains than they are cars. They're generally good at building things that go fast. But. Good. Ask them to build anything else. Not going to work. I build a Ferrari and I build on the pizza.  Yeah, you want to, you want a good tank.  All right. So these, can you see my page? Stream. So it looks like fastest is China. Second fastest is France. Third is Japan. Number four. Morocco. What? Morocco. Yeah. That was my question too. Morocco just gets its stuff from Europe, I think, but that's kind of weird because isn't like half of Morocco in open rebellion, right? Yeah. They're close to Spain. Like half of Morocco is like politics. Doesn't exist. I'm vaguely aware other countries even exist. It's me. I'm not even going to be aware of other states pretty soon. It's going to be me and my chickens. Living on fire. What's Virginia? I've never heard of this Virginia before. I only know my farm. Yeah, exactly. Okay. So Morocco and that's Spain below Morocco. You probably got the rail from Spain. You probably got the rail from Spain and took the safeties off of it. Made it go faster. Korea? Well, I'm surprised Korea is so far. And I'm actually surprised Germany is far behind. Germany is pretty car-centric too. Because of the Autobahn. Yeah, yeah. I guess you got a point there. See, another terrible thing that H-word did. H-word built the Autobahn or, you know, he had the dream for it and now everyone's driving their fancy German cars on the... on the Deutsch... on the Deutsch roads instead of... switching... instead of... getting in the trains responsibly, but he also, you know, he also did terrible things in the train. So I guess that's why Germans aren't allowed to build trains anymore. Okay. We got... let's see... talk about Epic Games... lawsuit against... Isn't that what you... what you texted me about earlier? I messaged you about E3 dying officially. Okay. Let's do that. You know what E3 is, right? I'll let you lead that topic. You've probably seen E3 before, right? Because that's been around for a while. Oh, okay, yes. Yes, got it. Okay, the video game thing. E3 Expo. So it's shut down. Yeah. You don't know what E3 is? It's a video game Expo, right? Yeah. It was like the... like it was a video game electronics Expo. So it's where you got like, not just video games, but like consoles and like... Right.  Yeah, I think I saw... I think I saw like maybe a video from Linus or DeBauer that were there. Yeah, it used to be pretty big. Like they used to have giant conventions in Europe and it's now officially dead. Officially dead, what happened? So it was kind of weird because like I think their previous record high was in 2009 or 10, like 700,000 total attendees over the... not 700,000, 70,000 total attendees. And then since then, they've been slowly climbing, mainly because developers, mainly Sony pulled out. I think in 20... Let me pull up the page here. I can back check myself. I'll do this yesterday, but... Sony pulled out a little while. Let me look at the thing here. When did Sony pull out of this? It's my hate by using CRX because I'm on a Belgian CRX server so I always get Belgian Wikipedia results. Yeah, so Sony opted out in 2019 because around 2019 they started doing this thing where they would have their own showcase.  And ever since then, like every company has started to kind of do that. And that was right before the pandemic. And when the pandemic happened, every company started doing their own showcase.  You said E3 did it online.    And then everybody pulled out. They couldn't get any more people, like developers to come there and they just died.  Like that's where you had the debut of like the 360 and the PlayStation 3 and the Wii. That's where all those got debuted originally. But now it's dead. And I don't really care. I'm not a huge...  I just kind of have to... I sometimes follow it because the other show I'm on talks about it a lot. So I tend to try to keep up. But I thought you would have known E3 because it started off in the 90s and I thought at least you would have probably seen it once or twice. That's what I'm saying. I think I watched either Der Bauer or Linus Sebastian at E3 like once or twice because I used to watch some of their videos a lot more. Actually, I don't remember if Der Bauer was active then. But I definitely watched Linus Tech Tip when I was in high school. Way more into computer hardware and stuff like that.  But at this point it's just like look up the specs, look up reviews of the thing when it comes out for my specific purposes. Which isn't even gaming anymore. I really don't play any video games anymore. So it's just like stuff that's more efficient for video editing and content production and stuff since that's what I'm doing, that's my business. Especially now that I have the farm I'm trying to take advantage of FarmTube. Make money as I'm making money by filming my day-to-day. That's like I really got red-pilled about that watching. There's the homesteaders but then there's other people like Al Blades the guy that cuts grass. You've probably seen some of his videos. Yeah, it's neither dude who cuts grass or people who just randomly for free. Yeah, so he'll cut usually it's abandoned houses but sometimes he'll actually do like a lawn restoration for somebody who who's not able to do it themselves or they're just lazy followers or whatever and he films himself doing it but I can tell you I don't know exactly what his what is it called RPM, his revenue per mile is but I mean I can tell you if it's similar to mine he's making bank off of some of these videos. This one here million views half an hour long he probably made easily $1,000 maybe $1,500 or more off of this these guys like some really good ones too popular 18 million views and it's an hour long probably I don't want to speculate too much about how much he makes again I don't have access to his RPM if you do the math if you assume that he makes $2 for every 1,000 views that's about what my RPM is but I also make shorter form content and the longer your content is the more you typically make assuming people actually watch it because there's more ads that can be ran so it's like if you just do that 18 million times it's 36 grand like very good for a one hour video and on top of that this guy pretty sure has a real landscaping business I don't think he's really advertising it too much here but there's a lot of people who do this like the girl with the dogs girl with the dogs she does a lot of free grooming for animals she made this 7 hours ago 117,000 views so she probably made a couple hundred bucks off it's a 5 minute video short so maybe not but I also suspect that animal videos might have higher RPM because A. the kinds of people who are watching them probably aren't using ad blockers and shit like that and B. the kinds of ads that are going to be run on these videos for like pet stuff I feel like there's more people that are actually going to click on those ads and buy stuff so yeah here she's advertising her store and I'm pretty sure she actually runs a legitimate grooming business as well she does these for free but she's able to make money through YouTube off of it another one that I like I think this guy is more of an underdog a meal plus subscribers is Victory Outdoor Services yeah I actually have more subs than this guy but he's pretty dope so this dude does um and he actually gets a lot of views on his stuff too which is good you know even though he's just got 4 in a K subscribers pretty much everyone is actually watching his videos um but yeah he runs like a concrete he does concrete he does um most of what I see him doing is concrete where he'll pour concrete for like a patio or things like that and it's like there's so many people who have businesses like this but you just put go pros on these people and you know you're shooting the shit you're having fun because like this is literally during COVID when I was doing work from home and stuff I would have like you know a spreadsheet or JIRA tickets and stuff like that open on one monitor and then I would have homesteaders open on another one you know people like this because I feel like for a lot of people doing work where you're just indoors all day long sitting at a computer is just not natural and so many of us want to be doing some kind of blue collar work but there's just not as much money in the blue collar work um but we want to live vicariously through these people which you can do the technology like you put you know go pros on like with this video that I edited or I gotta edit it together of me um showing off the coop and moving it but I'm hoping that it'll be the same kind of deal I mean I had four different camera angles and I could probably earn into a pretty decent video but yeah like this I really think that this is the future for businesses is to have some kind of social media presence and you can kind of double dip on your revenue here and make money through social media but also you're making money through a real job and honestly that's the kind of people that I think should be on social media in the first place spend so much time on Instagram or whatever or who spend time on YouTube trying to get famous and it's like you're getting famous for what? what's the point? you're some moron who's got an opinion just want to you know be famous off of that like it's so much better if you actually have something to sell some kind of product or service to sell and then social media to uh market that but anyway I think uh I think that's a podcast yeah thank you guys for watching this was the Libre podcast episode 5 great day


Hello, I welcome you all once again to my channel Explore Education. I am Dr. Rishmishy, Assistant Professor, Department of Education, S.S. Khanna Girls' Relic College, University of Allahabad. And after discussing the basics of statistics, this time I will discuss statistical series, Sankhikhi Shrankhalaay. And the lecture will be in bilingual mode and it must be interesting and useful for all of us. So, Arranging Classified Data in some logical order such as according to the time of occurrence, size or some other miserable or non-miserable characteristics is known as statistical series. Sorry, we are saying that what Sankhikhi Shrankhalaay means is that we have to arrange the classified data that we have. We have Vagikruth Akhda, we have to arrange them in some logical order, i.e. according to some logic, whether it can be the occurrence of time or according to the size or according to some other miserable or non-miserable characteristic, i.e. according to the time or according to the time of occurrence, if we arrange classified data in some logical order, then it is called statistical series. These series are prepared for the proper presentation of collected and classified data in an organized manner. And why are these series made? Because the data that we have collected and classified, we can present it in an organized manner, i.e. in an organized manner, properly in an organized manner, we can present it properly, that is why we make statistical series. So, you can get a basic classification of this, we can divide the statistical series into two groups, either according to their rules or according to their nature. So, according to the rules, statistical series or Sankhikhi Shrankhalaay can be of three types, time series, spatial series or condition series. So, time series, i.e. if the time series is based on time, then time series, if we have made a series based on space, then spatial series, if it is based on some condition, then it is a condition series. Similarly, if we divide them based on construction, then individual, discrete, oh, oh, oh, individual, discrete and continuous three types of things are kept. Individual, i.e. it is not a series, it is just some observations. Discrete means unsaturated, ungrouped frequency, i.e. we have not made any summary of the frequency distribution. And continuous series means grouped frequency distribution, i.e. we have made some groupings, then continuous, then of many types. So, now we know that time, spatial and condition are read based on the characteristic. What is time series? Based on the characteristic, it is the different values that are taken by this variable in a period arranged in chronological order. The series is called a time series. I.e. if we have made chronological order according to the period of time, i.e. we are looking at the birth date of Uttar Pradesh. So, in 1980, in 1990, in 2000, in 2010, in 2020 and in 2030, if we make it, then this will become a series and it is based on time. Based on a time period, then this will become a time series. Simply put, a time series is a statistical series in which the given data is presented in time units, i.e. day, month, week or year. In the same way, we can make the day, month, week as well, that every day, according to the month, according to the week, according to the year, i.e. the construction of the series is based on time. Similarly, spatial series, if the data is arranged according to the location and geographical consideration, then it is called a spatial series. i.e. if we are making it according to the period of time, according to the place, then the series will be like a time series. The time factor remains constant in this kind of statistical series and the place changes. For example, in 1980, Uttar Pradesh's tension is this much, Madhya Pradesh's tension is this much, Gujarat's tension is this much, Maharashtra's tension is this much and Chennai's tension is this much. So, the time factor is the same, because everyone is telling the 19th century but the place has changed. So, this spatial series will be like a condition series. If the data of a series is classified according to the changes occurring under certain conditions, then it is called a condition series. i.e. we have taken a place, we are talking about Uttar Pradesh's tension, we are talking about 1980. So, we have put some condition. For example, data is arranged based on the age of the student. i.e. based on age, based on the age of the student, we have put some condition like this. So, this will be a condition series. i.e. what is the characteristic of this? What are the rules? If the time is the main, then the time is the the time is the time. If the place is important, then the place is the the time is the time. The condition we have put the dasha, that is important, then it will be a condition series. Similarly, now it will come on the basis of construction. Sorry, I have lost it in my throat. So, this construction can be an individual series, it can be discreet and it can be continuous. So, individual means that it is just a matter of the eye. If it is unorganized, then it is unorganized individual series. If it has been organized, then it is organized individual series. There is no meaning in discreet. It is non-standard. There is no classification of it. But there are many continuous series. Continuous means Satatashankhla. Now, it can be exclusive, it can be inclusive, it can be open-end distribution, it can be cumulative. Look, this is my cut copy paste, so I am not able to make it right. This should be C U. Cumulative frequency series is equal and unequal class interval series and mid-value series. So, we will discuss some of them from you. So, now it comes on the basis of construction. We have an individual. Now, we are in an individual, organized individual and unorganized. Sorry, unorganized individual and organized individual. So, what is an individual series? The series in which the items are listed simply is known as individual series. That is, we have some tools, we have some items and we have kept it like this, without any grouping. So, it is individual. Even in that, it is unorganized. Series with raw data and unorganized, unarranged mass of data is known as an and, yes, in so many ways, there are words. A series with raw data and unarranged mass of data is known as an unorganized series. That is, it is raw data. We have just taken it. We have not classified it. We have not made any grouping. Just keep it like this. Raw data is the data. It is original form. That is, as soon as we have collected the data, it will be an unorganized individual series. Why? Because we have not made a grouping. So, the individual and the one who has got it, have not even organized it. So, unorganized. So, then the collected data will be raw. And then, when we organize the same data, not grouping, it will be individual. But if we organize it, it will be organized individual. A series with orderly arranged raw data is known as an organized individual series. So, there can be two ways to do it. There are two ways to present organized individual series. We can organize it in two ways. How? The first one is according to the serial number. Either we have only the serial number of the students of the class. So, first we had to check the copy. So, they are unorganized and keep the same number. But we have put the serial number from serial number 1 to serial number 50. So, this is an organized individual series. And either we can put it as per magnitude. Ascending or descending. Either the number of the child's copy is the first. And from the bottom. Or the one with the lowest number is the first and the lowest number is ascending or descending. You can use any method. So, arranging raw data in ascending or descending order of magnitude is known as an array, or arrays, or arraying of the data, or arraying of the figures. Array means to have a view. Something like this. View, crumb. So, this is it. Meaning, the raw data you got is an individual series. If you haven't organized it, it is unorganized. If you have organized it, it is organized. You can do it in two ways. You can do it simply in two ways. Either it comes from serial number or ascending or descending. Meaning, it increases or decreases. Which is called Arohi or Arohi crumb. Hey, tig. Why your discrete? What is discrete? If the number of observations of a study is large. If there are a lot of... I mean, we can make 50-100 children of the class as an individual series. 100 is more than that. But if there is a huge number of assumptions, then there is a huge gap. So, what do we have to do? It won't work with the individual series. What do we have to do? In that case, discrete series is a much better way to present the data. What do we have to do? We will make a discrete or ascended study. What do we have to do? In this frequency distribution series, different values of the variables are shown with their respective frequencies. What do we have to do? We have 10 to 15 to 20 children of the class. So, this is the frequency of respect. Meaning, Vargan Taral is talking about this. Then, the most important thing is the continuous series. Meaning, Satta Tashrankla. We have to read a lot of types of this. And the numerical we use, we also get continuous series. Explosive or inclusive. We will tell you later. A discrete series cannot take any value in interim. Meaning, the interval value does not give the discrete series. Therefore, in cases where it is essential to represent continuous variables with the range of values of different items of a given data, continuous series is used. Meaning, when we have made it discrete, we do not give any value in the interval that is coming in. That is not important for us. But when it is also important for the inside, then we have to make continuous series. It will not work with discrete series. So, in this series, the measurements are only approximations. And these approximations are expressed in the form of class interval. Meaning, Vargan Taral is important here. Other names of continuous series are. Its other names are like frequency distribution. Grouped frequency distribution. Series with class intervals and series of grouped data. That is, your frequency is divided according to that. Or, Avarati Vitaran is done. Or, Samuhi Avarati Vitaran is done. Or, Vargan Taral has Sarani. Or, Vargikrit Ankara has Sarani. These are a lot of names. In other words, a continuous series is one in which both the class interval and the corresponding class frequency are given. That is, the discrete series was only told to us by the frequency. That how many times it has been used. What does the frequency mean? How many times it is being used. But, the continuous series also tells us the frequency. That how many times it will be used. Avarati will also tell us. And Vargan Taral will also tell us. Okay. Now, you can see how continuous is done. That is all I have discussed with you. The most important is the exclusive series. And it works a lot. What is exclusive? Let's exclude. Let's exclude the upper limit. Look, it is a kind of continuous series in which the upper limit of any class interval is equal to the lower limit of the next class interval. For example, you must have asked me. Let's give it 0 to 10, 10 to 20, 20 to 30, 30 to 40. So, what is the upper limit of 0 to 10? 10. And what is the lower limit of 10 to 20? This is also 10. That is, the first one will count from 0 to 10. It will count from 0 to 9. It will count from the next 10.  That is, we are leaving the upper limit. So, an exclusive series is one in which every class interval excludes the value of its upper limit. That is, every Vargan Taral leaves its upper limit. The limit of its upper limit is exclusive. What is exclusive? We are taking everyone with us. Like, the series which includes all items includes the upper limit. So, how will Vargan Taral be included in this? From 0 to 9, 10 to 19, 20 to 29, 30 to 49. That is, from 0 to 9. We have everything. The upper one starts from 10. But what is exclusive? From 0 to 10, 10 to 20, 30 to 30. So, we consider ourselves as 10. Why did we leave 10? Because we have taken the lower limit of the upper one. Okay. Then the open end series is an open class. What will happen in this? In which the lower limit of the first class interval and the upper limit of the last class interval is missing. That is, the interval is also in this. But the first class will not get the lower limit. And the upper class will not get the upper limit. Open ended. Not closed ended. So, the first class will be less than 10. It is not saying that it is less than 10. Because it is less than 0, 1, 2, 3, 4, 5. It is not known. It is open ended. Similarly, the upper class is more than 50. From 50 to 50, it is not known. So, the first Vargan Taral, our lower limit is not known. It is missing. And the highest Vargan Taral, its upper limit is missing. So, it becomes an open end series. The question is asked less on this. Mostly, it is exclusive or inclusive. We get it for questioning. And what is the cumulative frequency series? What is cumulative? Yogatma. We are going together. For example, I have made a very long diagram. It is 0 to 10. So, in 0 to 10, it is 5 frequency. In 10 to 20, it is also 5. So, what will happen? 5 will be connected. It will be 10. What will happen in the upper class? And its frequency will be connected. That is, the upper or ascending, ascending, whatever order you are taking, the frequencies will be connected to the next Vargan Taral. Why is that? And the frequency series containing mid-values. What is in it? That type of avertisarani. In which the middle and middle are given. The middle and middle are given. So, which series, which have only mid-value of the class intervals and the corresponding frequencies. That is, it is not written as 0 to 10. It is written as 5. It is not written as 0 to 10. It is not written as 20 to 30. It is written as 15. That is, it is only given as mid-value and here it is given as corresponding frequency. So, this frequency series containing mid-values is done. So, you should know in one way which series you have got the question on.  These are the five main ones. That is, basically, we have to apply the length of the continuous series. So, this is the statistical series. So, if you know this, then we will ask questions on it. So, this has been the instruction of your teacher. And I have got this very much topic too. So, thank you and don't forget to like and subscribe to my channel Explore Education. I have done from this end.


All right cyber traders good morning and welcome on back and happy new year to everybody here live Great to see all of us here live inside the cyber group trading room And of course if you're joining us back on social media Facebook live or YouTube welcome on back folks great to see Michael Grant Rob Ray Phil Gary Lawrence welcome welcome back Lawrence one of the longtime members here Debra W. Doyle here as well Great to see everybody and again if you're joining us on social media welcome on back So new year new me right isn't that the case with everybody new year's resolutions new year new me and Not exactly the case right I know that we all have our new year's resolutions But it's the same strategy that we've been using that we will continue to use here across this calendar year And it's been working out pretty well so far at least this morning on this I MMX trade actually at first It was a bit of a dump-off and we'll talk about this trade to begin But now it's starting to pick up some momentum. So this stock ended up making a nice pop shortly after 8 a.m We ended up seeing news come out on the stock. I'm actually gonna pull it up right now for you folks All right, so I MMX shares trading higher after the company announced FDA granted rare Pediatric disease designation for their drug for the treatment of a life life threatening form of pediatric cancer in children All right, well it sounds like an incredible headline God bless right ended up leading to a big pop ended up topping off at first It's 650 but pulled back down. It was a nasty shake I was anticipating initially for this 580 level to act as support We had a lot of buying off the initial pop here So I was looking for this 580 level to hold beautifully a support It broke under and back above it on the same bar and then it rebroke it in tanks So right place wrong time look what's happening now though So it took a while and yes it ended up bottoming out right around five bucks If anything ended up beginning to hold a higher low from 525 here pushed up beautifully and right now We're getting that second wind type of move back over the 580 level so much So I ended up writing this in the chat board just a bit earlier folks And you know I even phrased it right away by saying I could easily be wrong on this exactly But you know we were watching this IMMX trade have a fight at that 580 level right before we started the meeting Actually about 10 minutes prior and I told you folks I said hey, you know I could be wrong on this But I think after 9 a.m. Eastern time we will see the IMMX Make a make the bigger pop off the 580 level and then nonetheless here we are now So I hope that you folks were able to jump in right from the 580 if not earlier perhaps off 525 And for right now the way this stock looks it's incredible So no better way to start the new year off than the nice runner on IMMX All right I should have started off this whole meeting for all of us in social media at least of course you could tell Across the first three to five minutes. This is not the voice of voice of one Fausto Puglisi My name is Josh Levitin. I'm Fausto senior instructor at Cybertrading University I know the Fausto had mentioned just back on Friday that he would be out for this week He's in Florida. He's actually shooting a commercial for CTO a couple of other business related things going on down there So he'll be back and for all of us here live inside the cyber group room little housekeeping note And we talked about this last week as well This week we have the phase 3 stock course. I believe that will be taking place Wednesday and Thursday this week No Fausto. He got me for phase 3 folks So for Tony for Grant for Mark for all of us here that are part of the phase 3 stock course At least you're gonna see me trade live on Wednesday and Thursday. All right So what we'll do just to kind of make things simple because truth be told, you know rolling into the new year I forgot what I had for breakfast this morning, let alone the stocks be traded back on Friday So we'll go right from top to bottom We'll go across the gainers and losers lists here and see what else we could add to our watch list alongside the IMMX trade How about this one of us in social media on YouTube? They got in on the IMMX. They just posted earlier They jumped in at 527 nice entry cool and Mark marks in in cyber group right now from 517. He has a really good entry on that IMMX trade congrats Mark and Funny name on YouTube sleepy fishing bear. I like that name there. I'll pop that up on the YouTube stream All right, so let's keep it going folks. Let's see what else we got here going into this morning Otherwise aside from the IMMX pretty much every other stock on the pre-market list Mind you pretty much look like this CELZ not doing a whole lot right now ended up making a nice pop from the from the early pre-market this morning It's still very active in the sense of volume traded across the pre-market session It has liquid order book and all the good stuff that we would want to trade So I of course will put this on our list. Hopefully this shows a lot more signs of life prior to 9 30 coming up here Stocks trending down. So when you talk about a stock trending down leading into the open, you know That's typically driven by selling so history repeats itself This is only going to continue to dump dump off So we want to make sure that we see some signs of life prior to the market opening up here All right, let's keep it going AFI very similar right another stock This was actually from Friday afternoon that ended up leading to a nice after hours pop What was the news on this one at least AFI the company says it's received 35 million dollar capital infusion I don't know what that means Announces process for sale of the company and consideration of other strategic alternatives I'm sure there's a lot more to it aside from that fairly generic headline But hey for right now very similar to the CELZ trade just a lot of activity throughout the pre-market this one Seems like it's trying to show some signs of life You know after 845 Eastern time bit of a pop some buying came came in roughly off of 270 It needs to break back over that 270 trade here folks, but alongside the others I'll put the AFI on our list here still All right, so we keep it going we keep it moving BLCT another one right and it's just slow mover here This one trying to eke up you know bounce from 180 up to 190 major resistance you would assume to be off of 2 and Otherwise same line of thinking we ended up trading this one back on Friday as well like the AFI But ended up leading to a nice pop You know topped off back Friday like right around 170 at the time 165 Right here So you want to make sure I could hold above that price at least but nonetheless the BLCT will put on our list as well Not a bad list so far the IMX is definitely the most interesting of the bunch We're not going to trade Tesla folks Just wanted to show it here pretty quickly at least I post the news for anyone that has any you know calls on it Nonetheless, but uh, you know big move for Tesla this morning up over seven and a half percent They're trading higher after they ended up reporting better than expected vehicle production and deliveries in the fourth quarter So just wanted to give you the insight at least on Tesla. We're not going to trade us $1,100 stock though PTP I little push up I know one of us had called this out earlier and he was trying to go up It's up over seven percent right now and trading over 500,000 shares So it would be hard for us to not put this on our list and then also on the flip on the same side of that At least we are pretty familiar with this PTP I back over the last, you know, really month or so in full back throughout December We were trading the stock on and off So, you know, I'm sure on a day or two. It was uneventful I'm sure on a day or two is pretty nasty, but nonetheless at least we know this stock is pretty liquid and it's cheap It's one that we're familiar with at least so Again, I'll put this on our list for right now I'll tell you what with Tesla popping I should have mentioned this just before we did call out a pop on Neo Which is another EV company, you know Chinese EV company And I don't know if it's directly related to Tesla if it's like a sympathy play for that matter. Let's see Oh, no, it's news of their own actually and very similar news to be honest So Neo shares trading higher after the company announced December vehicle deliveries increased 50% from the previous year So same line of thinking as Tesla, but this one much more within our price range Definitely gonna put the Neo on our watch list. We'll check out some of the other EV stocks like Nikola, which, you know, not a Really a great trade really right now XPEV gapped up in pre-market little expensive lower highs it's building So we don't need to put every EV stock on our list here folks solo I'm sure I'm missing one or two, but nonetheless the Neo trade looks really good at least NIO All right, let's see if there's anything else that we could add to our list here Otherwise just to be transparent looking at the volume throughout a lot of these stocks in pre-market kind of light One of us calling out GNPX, which actually rings the bell right away to me I'll look at that pretty quickly then we'll jump over to the losers list see if there's anything looking just as good if not better Yeah, gmpx. Wow, did I skip over this or just not pop up on my scanner here? Popped up on my custom scan here, but the main gainers list I don't know why it's not showing perhaps a glitch on the trade station gainers side there So yeah, I appreciate that for one of us on YouTube again a sleepy fishing bear I like that name stick with that name, but nonetheless, this is a great stock definitely gonna put this on our list and this stock rings a bell to myself It should ring a bell to a few of us at least back You know if you were in the trading room back in March of 2020 or January actually this was this was the first stock the first stock back in January of 2020 that popped off of some sort of potential COVID vaccine news This was the first one after this there is CODX There is obviously the more expensive ones that ended up taking off, but this was number one I remember that like the back of my hand so You know things like that you'll always seem to kind of remember at least you know global pandemic You know all those stocks we ended up trading. This was the first so right now. It's moving up pretty nicely I like this stock and check to see if there is any news on this trade All right, so there is news of course not related to COVID GMP shares trading higher after the company announced it received FDA fast-track designation Incombing or they're treating for non non small cell lung cancer That's the news on it. They're drug that I can't really pronounce But they're treating non small cell lung cancer is the news on this so Cool, I like it. Good call. We'll hop over to the losers list right now seeing if there's anything just as good if not better I'll tell you what this APLT I could have put on the pre-market list at first, you know prior to 8 a.m This thing looks like trash now, but it gapped down going into the pre-market and made a nice, you know squeeze up I caught this much later on as it was pulling back here I didn't even call it out in the trading room at the time because I think at that point the IMMX ended up taking our attention So I don't know this thing's too flat right now, but it had a lot of interesting activity early in the pre-market I don't know what the news is. Let's just check might be common stock offering or some sort of stuff No, actually on the flip side. We were just talking about GMPX getting FDA fast-track approval Looks like the FDA ended up putting a hold on APLT's drug to treat rare metabolic disorder So, I don't know. It's just really flat right now. I'm not a fan of this So I'm gonna move on from this stock for right now the news that came out the big, you know Dead cat bounce and that was pretty interesting early on All right, let's see what else we got here folks. I know that that IMMX trade ended up completely popping So I'm very happy to hear that we'll jump back that stock at the end KPRX familiar name from last week not doing a whole lot at least on the upside What about NTRB from NTRB from mark It's just a straight dart on the way down ended up making a really nice move yet Friday going into the morning But it was trading really thin volume Friday, too So, you know, this is where we can generalize and say what makes the top percent gainers list one day Typically makes the percent losers list the following. So, you know, hey, it's a familiar name It's one that we did really well on from Friday But the way this is moving so far it only seems to me as if it's just gonna keep dumping off I wouldn't mind calling the stock out after the open mark once we happen to see maybe some sort of support look to build But I feel like this is just gonna completely come back down at first Look at that pop IMMX Right once it broke back over that 580 that that's all it took folks 580 was the big buying level that we had early in the pre market That's why I kept typing that number in the chat board to you folks. All right. Oh really quick also an IMMX I noticed this earlier. I also typed this in the chat board This is all we got on the daily chart. This is a new IPO, I guess so already breaking all-time highs I think the previous high back from late December 631 so obviously above that price right now. This looks really good No historical resistance to work off of from the daily chart at least as of now All right, folks, I don't think I need to help here I think that we're pretty much good to go at this point, but I will of course always ask Let me know if there's anything up down left right more than 5% that I may have missed I think that we touched upon them all Wouldn't mind being wrong if there is one more that popped that I didn't look at just yet Dickens in from 555 out at 657 right there on this IMMX trade live in Cybergurt Excellent way to start off the New Year that New Year there dickens All right, folks going once going twice Going three times and that is it. I think that we're pretty much good Listen, if there is anything else that makes a big pop or perhaps even a big drop You know going into the open obviously will look to call it out inside the Cybergurt room at least But I'll tell you we have a good list to start. We don't we have six stocks to work with Not too big not too small. I think the shoe just fits all right folks So aside from that again if there is anything else we'll look to work together as a team inside our live trading room and call it out When the time is right. All right, so for a sleepy fishing bear and for all of us on YouTube for all of us on Facebook I appreciate you sticking with me each and every day each and every morning perhaps now But otherwise again, I'll be doing the morning meeting for foul So I believe across this entire week if you have any questions You have a couple of options for all of us in social media You could always send over a quick email for all of us that don't join me in the afternoons at 2 30 Eastern time I always like to post my email address at the end of the meetings that I do So if you have any questions any questions about our call outs or watch lists Feel more than free to send a question right over to my email address Josh at ctu trading calm Otherwise if you wanted to continue the audio commentary in our live trading room You could just go right down to the link on the banner below to get yourself started All right, so for Michael for Grant for Mark for Dickens help a trade their Dickens as well as Lawrence Tony Susan Deborah all of us live inside Cybergurt happy new year I look forward to this year this year coming up folks not just in terms of prosperity But more importantly much more importantly nowadays in terms of health All right So wishing everybody the best in health and the happiest of New Year's ahead But again, we'll look to continue the audio broadcast here coming up in just about 15 or so minutes. Take care


Okay, let's do some Python on hardware. Yay, it's blanket time. It's blanket time. First up, Morgan Stanley has a Morgan Stanley Makerspace program, and we post it up in our latest newsletter. They have a circuit playground, what circuit Python thing you're doing, so lots of kids getting hardware able to fill their Makerspaces with electronics so they can learn how to code, check it out. Colorful LEDs, look at it, it looks so cool. And if you want to check it out, they're not just putting this on a separate Twitter account. This is on twitter.com slash Morgan Stanley. So you know it's a big deal when the big companies are doing stuff with kids and code and Makerspaces, and they use their main account, not like the name of the bank or whatever it like gives. No, they're fully behind this. So very cool to see Morgan Stanley doing this. Next up, the latest issue of Hackspace Magazine is out. We have a lot of the articles posted on our site that you can go and read. Don't forget to subscribe. If you subscribe, you can go and use the Hackspace mag.cc slash Adafruit and subscribe. But it's Python on hardware. And that's one of the big articles in there. Drew wrote it from Osh Park. And he loves it because it's like microcontrollers, microcomputers, everything is supported. And they definitely have a lot of Python code and projects in Hackspace every month. It's a monthly magazine. So subscribe. They're coming to the US. So check that out. And in addition to all the Python on hardware stuff in there and using Python to control LEDs, there's Design of Vos with Fusion 360, makes things with toothpicks, add audio to Raspberry Pi projects, and 3D print strong parts. Next up, this is kind of neat. If you've ever wanted to get the measurements out of a caliper and use them in some way, either to store them or do something with it with an external device, CircuitPython measures up. There's a CircuitPython library for parsing data out of a two-toilet instrument like calipers or micrometers. This is great especially if you're doing a lot of automated measurements. You don't want to have one person recording and one person measuring. You can just have it all happen automatically for your calipers. During Engineer's Week, DigiKey did a really neat thing. They had, and I'll just show the video, and we'll just talk over it. They had a girls' night out for Engineering Week. And this year, they hosted one for all the young women in their area to increase interest in STEM. They learned how to program a Circuit Playground from Adafruit, the test. They used make code. They used CircuitPython. They used things to build projects that they can keep for later. And the staff helped out. So this is a really good example of one of our partners doing something that scales really. For us, we can't really do workshops in our space and all that, but this is something they did. So this is really neat to see. So thank you, DigiKey, for being such a great partner and very cool work. And look at all these projects that they were able to do. I love how all these girls, they have their own interests and backgrounds and experiences. And so when they become interested in electronics or engineering, they're going to come with their own information and background. And it won't just be like what they learned in school. It'll be like these kinds of events that give them inspiration to design new and interesting electronics. Next, on Hackaday, they had new part day. And oh, the things you can do with a clue. So check that over on Hackaday. They have a new part feature and they had the clue on there. Next step, over on Oshpark, there is a post about CircuitBrain Deluxe, which packs CircuitPython into a one inch square. Yeah, handy. Some other things that are happening. This is something we've been noticing. This is a tweet from Isaac Kelly. My first Hello World in maybe a decade, but maybe I'm finally around, finally getting around to playing with CircuitPython. It's impressively straightforward and definitely the most pleasant code on microcontroller experience they had. That's what we were going for. We don't know this person. They just tweeted it out. We saw it, we thought, this is neat because that's what we like to hear. Even if you haven't been doing any type of coding or Hello World in 10 years, if you could do something this easy, this fast, super cool. Also, this is maybe a more advanced engineer. Maybe someone has been doing this. And this is from Jordan. Most of my firmware flashing experience is requiring specialized software and off of serial cable. It's frustrating an airplane. For CircuitPython, you plug it in via USB and it shows up as disk. You copy a file to a flash, to flash the new firmware. This is so simple and good. I have like six different DFU upload programs on my Windows computer, and they're all completely incompatible. Yeah. I listen to LeMore's computer all the time and it goes, bada, bada, bada, bada. That's all it does. Bada, bada, bada, bada. Then you have to reboot sometimes. I know. This is kind of cool. Cedar Grove is learning how to use the CircuitPython display I.O. First to make text and labels, then an animated battery monitor, Sprite, and an automatic DST for the attached RTC real-time clock. This is a nice little display. Yeah. This is kind of neat, bright wearables, this is a geek mom. It makes wearables and things that happen to use. Anywho, geek mom has wearable clues and it's all working out now. So this is the clue that goes into the wearable of the Gripboard, and it was a micro bit. Yeah, you get that RGB LED is pointing at the color sensor, and so she's actually changing the NeoPixel based on the color sensor built into the clue, which is neat, because there's all these sensors that the clue has that the micro bit is not. You can do more advanced projects. Okay, next up, you can check out all the new feathers. Lydia, maybe you can talk about this. This is the upcoming super early Don't Ask, it's not out yet. Feather M7, RT-1011, designed by Arturo168, I can't remember the number, his name, or 182, who's been doing a lot with the IMX NXP chip set. So this is a very inexpensive but very powerful chip. It was at 500 megahertz, and we thought, hey, wouldn't it be cool to make this into a feather? This is very, very fast, and we can, when Circuit Python on it, we already have a build of Circuit Python and working on having the U of T bootloader as well, but basically what we wanted to do is try to get this $1 chip onto a feather so we could get it onto the world, because I think a lot more people would use this chip if it was available, and it's coming soon, soon, soon. But very exciting. Four layer board with a USB-C connector. Okay, and this is one of the new learn guides. This is a new guide from Brian. This is the STM32F405 Feather running Circuit Python with an OLED display and a pressure sensor with a port on it, and the port has a tubing connected to it, and so you can turn it into a SIP and PUF sensor. And what's neat is that Brian did a really interesting structure where there's a callback system for different SIP and PUF strengths. So if you have a weak SIP or a strong PUF, you'll get a callback and then you can form different functions based on what the pressure sensor measured. So this is for making alternative user interfaces, sometimes for accessibility, but also, like, Phil, you were thinking, what if you had a puppet that you could control some of it using your mouth, that there was that musician who would have a tube that wouldn't use a guitar? Can't remember the name of him off the top of my head, but so you can use the pressure of your mouth to, as a sensor, to read and control other mechatronics, robotics, what have you. Okay, and then just a little bit of a, do we have 214 libraries? So many. Circuit Python. And even more contributed libraries. And don't forget, coming up pretty soon, the Open Hardware Summit is in March. We have Circuit Python on the wearable watch badge. Correct. Made by Osh Park, SparkFun, DigiKey, and I think each participant will get one. Yeah. We also will have 3,000 clues at PyCon. You'll get a clue there. If you're at PyCon, we will have a clue for you. And that is Python on hardware news for this week. Python on joysticks. Yeah.


Assalamu alaikum. Welcome to lecture number 16 of the course on statistics and probability. Students, today is the 16th lecture and the beginning of the second part of this course. As I mentioned in the very first lecture, the first 15 or so lectures were concerned with descriptive statistics. The next 15 are going to be concerned with probability theory and the last 15 will pertain to inferential statistics. So, during the first 15 lectures, you have learnt different techniques of summarizing and describing data, data that you collect on sample basis. Our general discussion was about the univariate situation in which we were describing a single variable in different ways. And in case of a quantitative variable, we discussed at length the frequency distribution of the variable and its shape, its spread and its measure of central tendency. Also in the last lecture, I discussed with you a very important concept and that was regression and correlation when we tried to relate two variables with each other. I hope that you enjoyed learning the various concepts that were discussed in the first part of the course and I hope that you will enjoy probability theory as much if not more. You will say probability as a difficult concept we can enjoy. Actually students, this is not the case. This is an impression that is all around that it is supposed to be the most difficult topic. The only thing is that you need to have a methodological approach. If you approach it in a systematic way, then you will see that it is not really that difficult and in fact it is quite enjoyable. In this part of the course, we will begin with the basic concepts of probability and we will go on to discuss discrete and continuous probability distributions. In particular, we will be discussing the binomial distribution, the hypergeometric and the Poisson distributions and also the most important distribution in statistical theory, the normal distribution. So, let us begin with the very basics. First of all, let us discuss the definition of probability. What do you understand by the word probability? I am sure that you will reply that when we are talking about chance, that is what we mean by probability. We are talking about the chance of something, the chance of something. But in the next few lectures, you will see that there are various ways in which you can define probability. We have the classical definition, we have the relative frequency definition and then we also have the subjective or the personalistic definition of probability. These are definitions, we will discuss one after another and you will realize that the most important one from the statistical point of view is not the subjective, not even the classical, but the relative frequency definition of probability. Because this is the definition that pertains to real life phenomena and also a definition which enables you to quantify probability. You can express it in a number form, for example, a particular event probability is 75 percent and another event probability is 55 percent. We will discuss these things in detail during these 15 lectures, but today we have to start from the very basics. Students, the first thing that we will discuss in this regard is set theory. It will simply be a review of what you have already studied and the reason why we are going to discuss it is that as you will see later, there are a number of concepts of probability theory which are facilitated by the use of set theory. Let us start from the definition of a set. As you all know, a set is a well-defined collection or list of distinct objects. For example, a group of students, the books in a library, the integers between 1 and 100, all human beings on earth, etc. etc. In its definition, I use two words, well-defined collection of objects and distinct objects. What do these two mean? Well-defined means that I should be absolutely clear as to whether a particular object belongs to or not belongs to a particular set or distinct se yehi murad hai that any particular element or object should appear in that set once and only once. Students, the set itself is denoted by a capital letter usually such as capital A, capital B, capital C and the elements in the set are denoted by small letters. As you see in the examples on the screen, the set A could be consisting of small a, small b, small c and small d and the set capital B may consist of the numbers 1, 2, 3 and 7. The number of elements in a set A, this is denoted by n A and it is called the number of the set A. As you see on the screen, if a set consists of 4 elements then we will say that the number of this set is 4 and we will write n A is equal to 4. If x is an element of a set A, we write x belongs to A and if x does not belong to A then we write it with the same notation but crossing on the notation to denote that x does not belong to A. As you can appreciate a set can have any number of elements and if a set does not have even a one single element it is called an empty set or a null set and it is denoted by the Greek letter phi. But students please note that the set consisting of the element 0 is not a null set because it contains one element and that is 0. In fact, if a set contains only one element we say that it is a unit set or a singleton set. But students, if we are talking about an element x then we will not put a parenthesis or bracket but if we are talking about a set containing one element x then we will be putting the parenthesis around the letter x as you now see on the screen. A set may be specified in two ways the first is called the roster method and in this method we give a list of all the elements of a set. For example, if we throw a die our set of all possible outcomes consists of 1, 2, 3, 4, 5 and 6 and if we toss a coin our set consists of head and tail. The other method is called the set builder method or the rule method and in this method we state a rule that enables us to determine whether or not a given object is a member of our set. For example, if we write that a is the set of all x values such that x is an odd number and x is less than 12 this means that we are talking about the set 1, 3, 5, 7, 9 and 11. Speaking of sets it should be kept in mind that the repetition of an element or the change of order of the elements in a set does not alter the set meaning that if I write 1, 3, 5 that is the same set as 5, 3, 1 or 5, 1, 3. The size of a set is given by the number of elements present in it. The number NA in other words denotes the size of a set. This number may be finite or infinite. A set is finite when it contains a finite number of elements otherwise it is called an infinite set. The empty set is regarded as a finite set. Examples of finite sets would be the set A of consisting of all the positive integers from 1 to 100. The set B consisting of x values where x represents a month of the year, C representing those x values which represent printing mistakes in a book and D representing those x values which represent living citizens of Pakistan. In the examples may agree that we are talking about finite sets because number of living citizens of Pakistan is not an infinite number. Number of mistakes in a book cannot be an infinite number and so on. On the other hand examples of infinite sets are the set of even integers or the set of all real numbers between 0 and 1 including 0 and 1 or the set of points on a line or the set of the sentences in the English language. Of course it can be argued that the number of sentences in the English language is not an infinite number but the point to understand is that sometimes if the set or if a population is very very large so large that for practical purposes we can regard it as infinite then we do adopt this strategy. On the other hand in the first three examples it is obvious that they are really infinite sets because the total number of even integers or the number of real numbers all the real numbers between 0 and 1 or the number of points on a line they cannot be regarded as finite entities. These four examples that I have just given you will note that the first three examples were the set of all points on a line or the set of even integers or the set of all numbers between 0 and 1 inclusive because they you can simply not count them as finite entities. But the last example the set of all sentences of the English language it cannot be an infinite number truly speaking. So, sometimes when sets or samples or populations if they are so large that for all practical purposes they can be regarded as infinite then we do adopt this concept. A set that consists of some elements of another set is called a subset of that set. If B is a subset of A then every member of set B is also a member of set A. For example if set A consists of the elements 1 2 3 4 5 and 10 and B consists of 1 3 and 5 then B is a subset of A in other words B is contained in A. Speaking of subsets it should be noted that every set is regarded as a subset of itself and the null set phi is regarded as a subset of every set. Two sets A and B are equal or identical if and only if they contain exactly the same elements. In other words set A is equal to set B if and only if A is a subset of B and B is a subset of A. Speaking of subsets we also need to differentiate between the proper subset and the improper subset. As you see on the screen if a set B contains some but not all of the elements of another set A while A contains each element of B. In other words if B is contained in A and B is unequal to A then the set B is defined to be a proper subset of A. A very important concept is that of the universal set that set of which all other sets are subsets is called the universal set. As you now see on the screen it is the set which contains all possible elements under consideration. It is also called the space and it is denoted by either by capital S or by capital omega. An interesting question is how many subsets can a set have? As you see on the screen a set S containing n elements will produce a totality of 2 raised to n subsets including the set S and the null set phi. For example, if we consider the set A consisting of the numbers 1, 2 and 3 this set will contain it will generate 2 raised to 3 that is 8 subsets and these subsets are phi 1, 2, 3, 1, 2, 1, 3, 2, 3 and 1, 2, 3. In the context of set theory a very important and useful concept is that of the Venn diagram. A diagram that represents sets by circular regions parts of circular regions or their compliments with respect to a rectangle representing the space S is called a Venn diagram. The Venn diagrams are used to represent sets and subsets in a pictorial way and to verify the relationship among the sets and the subsets. A simple Venn diagram is of the form that you now see on the screen and the one that you see at this time pertains to disjoint sets those two which do not have any element in common. On the contrary if there are two sets A and B such that they have a few elements in common then the Venn diagram is of the form that you now see on the screen. The next concept that we must concentrate on is the concept of operations that can be performed on sets. Sets can be combined in order to form new sets and the four main operations that we should consider are union, intersection, complimentation and set difference. The union or sum of two sets A and B means the set of all elements that belong to at least one of the sets A and B that is A union B is the set of all those x values which belong either to A or to B. For example, if set A consists of the numbers 1, 2, 3, 4 while set B consists of 3, 4, 5, 6 then A union B consists of the numbers 1, 2, 3, 4, 5 and 6 or the new set 1, 2, 3, 4, 5 and 6 students note that it is fulfilling the basic property that I just mentioned in terms of union. So, some of them belong to A but not to B, some of them belong to B but not to A and two of them belong to both A and B. So, we can say that the elements of this set belong to at least one of the two basic sets A and B. The intersection of two sets A and B means that set in which the elements belong to both A and B. For example, if set A consists of 1, 2, 3, 4 and set B consists of 3, 4, 5 and 6 then A intersection B is the set of the elements 3 and 4 as these are the two elements which belong to both A and B. Two sets A and B are defined to be disjoint or mutually exclusive or non-overlapping when they have no elements in common that is the intersection of the two sets is the null set 5. For example, if set A consists of all possible outcomes of a die that is 1, 2, 3, 4, 5 and 6 and set B consists of all possible outcomes when we throw a coin that is head and tail then students it is obvious that these are disjoint sets in the sense that they have no element in common. The difference of two sets A and B denoted by A minus B is the set of all elements of A which do not belong to B. Representing this point by the Venn diagram that you now see on the screen, the shaded area of the set A represents the set A minus B because it is all those elements of A which do not belong to B. And the last concept in this regard is that of complementation. The particular difference S minus A that is the set of all those elements of S which do not belong to A is called the complement of A and it is denoted either by A bar or by A C. Symbolically A bar is the set of all those x values which belong to S but which do not belong to A. In the Venn diagram that you now see the shaded portion of the universal set S represents the complement of A because it contains all those elements which are not contained in A. The next important point to be considered is the algebra of sets and this is that part of set theory which provides us with a number of laws that enable us to solve a number of problems. The first law is the commutative law and as you now see on the screen it is given by A union B is equal to B union A and A intersection B is equal to B intersection A. I think that you will agree that this law is quite self-evident. Zahir hai ke aapne A aur B ka agar union lena hai to aapne unko hi kathya karna hai to agar aap A union B kaheen ya B union A kaheen it is one and the same thing and the same holds for intersection. The next law is the associative law and it states that A union C is the same thing as A union B union C and similarly for intersection. The distributive law is given by A intersection B union C is equal to A intersection B union A intersection C. Also A union B intersection C is equal to A union B intersection A union C. Students iss law ko yaad rakhne ka aasan tari kaya hai ke aap ordinary multiplication or addition ka example apne zaheen merakheen. When we write A into B plus C we know that it is equal to A into B plus A into C. Isi tara se agar aap isko dekhenge to you will see that it is exactly the same pattern. The idempotent law is extremely obvious A union A is equal to A and A intersection A is also equal to A. The identity laws are A union S is equal to S A intersection S is equal to A. A union phi is A and A intersection phi is equal to phi. I would like to encourage you to draw the Venn diagram and to work with it yourself and to decide for yourself whether or not these are correct. Similarly, we have some other laws as you now see on the screen. The complementation laws are given by A union A bar is equal to S A intersection A bar is equal to phi. A double bar is equal to A S bar is equal to phi and phi bar is equal to S. Also we have the D Morgan's laws and these are given by A union B whole bar is equal to A bar intersection B bar and A intersection B whole bar is equal to A bar union B bar. Once again it is very simple to verify all these laws. Aap kuch numerical examples le lije take examples of sets which contain various numbers and then verify each and every one of these laws. The next concept is that of the partition of a set. And as you now see on the screen, a partition of a set S is a subdivision of the set into non-empty subsets that are disjoint and exhaustive that is their union is the set S itself. This implies that a i intersection a j is equal to phi where i is unequal to j and a 1 union a 2 union a 3 so on up to a n is equal to S. Let me explain this to you with the help of an example. Consider the set of all possible outcomes when we throw a die and that is 1 2 3 4 5 and 6. Now, if I partition this set into three parts in such a way that the first part consists of the elements 1 2, the second part consists of 3 4 and the last part consists of 5 6. Students this is a partition of the set S into three parts and it fulfills the formal definition that I just presented to you. The union of the three sets 1 2 3 4 and 5 6 is equal to the big set 1 2 3 4 5 and every one of these three sets is mutually exclusive 1 2 is different from 3 4 and 3 4 is different from 5 6 and there is no part of any of these three sets which is overlapping with any other set. The next concept is that of the class of sets. By a class of sets I mean a set of sets. For example, if you consider the set of lines this is a class of sets because each line itself is a set of points. The class of all subsets of a set A is called the power set of A and it is denoted by P of A. For example, if A is the set head and tail then the power set is given by the null set phi, the set H, the set T and the set HT. You will remember that a few minutes ago I told you that if a set consists of N elements then the total number of subsets that it can produce is 2 raised to N and we also considered an example there. So, the definition given under that example, the set that we considered of the 2 raised to N subsets of the set that set was also a power set. Another very important and interesting concept is that of the Cartesian product of sets. As you now see on the screen the Cartesian product of sets A and B denoted by A cross B is a set that contains all ordered pairs x y where x belongs to A and y belongs to B. For example, if the set A consists of HT and the set B consists of 1, 2, 3, 4, 5 and 6 then the Cartesian product A cross B is the set of the ordered pairs H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5 and T6. This silsle may important point here is that in general A cross B is not the same thing as B cross A because if you consider the graph in which x is on the x axis and y is on the y axis you will readily agree that 1 comma 2 is not the same point as 2 comma 1. Students I have discussed with you the basic concepts of set theory which I am sure that many of you are already familiar with, but it is good to revise and to refresh them in your mind because as you will see in the forthcoming lectures this theory will be very, very useful for us when we talk about probability and a number of probabilistic problems. Another important mathematical theory which we will be using in solving probabilistic problems is the theory of counting rules. There are three rules that enable us to solve a number of problems in a convenient manner and these are the rule of multiplication, the rule of permutations and the rule of combinations and I will pick them up one by one. As you now see on the screen the rule of multiplication is stated as follows. If a compound experiment consists of two experiments such that the first experiment has exactly M distinct outcomes and corresponding to each outcome of the first experiment there can be n distinct outcomes of the second one then the compound experiment has exactly M n outcomes. Students, this rule is extremely simple. Let me explain it to you with this rule. With the help of an example, let us go back to exactly the same one that we discussed a short while ago. One set consisting of h t and the other one consisting of 1, 2, 3, 4, 5 and 6. I am sure that this is the case of tossing a coin as well as tossing a die. The coin can result in two possible outcomes head and tail and so M is equal to 2 and the die can result in six possible outcomes and so n is equal to 6. Now according to the multiplication rule the total number of ways in which this compound experiment can be performed is M into n. So, in this case it is 2 into 6 and that is 12 and students as you will remember 12 is exactly the number of ordered pairs that we had when we considered this example earlier as you now see on the screen. The 12 ordered pairs are H 1, H 2, H 3, H 4, H 5, H 6, T 1, T 2, T 3, T 4, T 5 and T 6. Here is the example I have just discussed. This may be considered two simple experiments being performed together to give us the compound experiment, but students this can be extended to any number of simple experiments which are being conducted together to give us the compound experiment. And the other point is that this rule of multiplication theorem is also called the rule of multiple choice and the example that I will now consider and discuss with you will portray this point clearly. As you now see on the screen suppose that a restaurant offers three types of soups, four types of sandwiches and two types of desserts. Then a customer can order any one out of 3 into 4 into 2 that is 24 different meals. Let us consider the points that I just conveyed to you. This has three simple experiments. The choosing of one soup out of the three possible choices, the choosing of one sandwich out of the four possible sandwiches and the choosing of one dessert out of two. Why this rule is also called the rule of multiple choice. The choosing of the soup, the choosing of the sandwich and the choosing of the dessert. So we are extending the initial rule M n to the case of three experiments and we are now applying the rule M into N into P where M is the number of ways of choosing a soup, N is the number of ways of choosing a sandwich and P is the number of ways of choosing a dessert. Let us consider another very interesting example. Suppose that we have a combination lock on which there are eight rings. In how many ways can the lock be adjusted? This lock has eight positions and we are trying to determine how many different ways in which we can adjust or set this lock in order to lock the suitcase. Now students, how do we approach this problem? The logical way to go about it is to think of the eight positions and to realize that for any one of those eight positions it can be filled in ten different ways. There are ten digits from 0 to 9 or juphehli position hai uspe bhi 0 to 9 koi bhi digit hum rakh sakte hai. Same for the second position, same for the third and same for the eighth. So if we apply the rule of multiplication, we find as you now see on the screen that the total number of ways of doing this is ten into ten into ten and so on, so that the answer is one hundred million. Can you imagine one hundred million ways of adjusting that lock on your suitcase? One hundred million yani das karoor and the formula in this case when all eight positions have the same number of possible ways of being set, the formula can be simply stated as ten raised to eight, ten into ten into ten eight times. But fundamentally it is the same rule that I have been discussing for the past few minutes, the rule of multiplication. Students this brings us to the end of today's lecture and in the next lecture I will discuss with you the other two very important rules which enable us to solve a number of problems in probability theory, the rule of permutation and the rule of combination. After that we will proceed to the real thing itself and that is probability. Until next time, my best wishes to you and Allah Hafiz.