Random Variables || Discrete and Continuous Random Variables

Added: 2026-05-05

[00:04:38]All right. Uh, welcome everyone.

[00:04:43]As I promised yesterday, I want to do a little recap on the introduction to random variables.

[00:05:23]All right. So, uh please pay attention. Let me give you the real uh the real uh understanding of random variables in case you missed out on yesterday's class. Now uh in our previous classes on probability I believe every student in this class is familiar with outcomes of random processes. When we say random processes I'm sure you understand that we're talking about processes that is the the processes in which their outcomes are by chance. There is no predefined result.

[00:06:01]For instance, if you pick a coin up and toss it up, you don't have uh any assurance of having probably a head or a tail. It's all by chance. If you throw a coin up right now, it could give you a head. If you if you toss it up the second time, it could give you a tail. That's a very clear example of the random process. Now another example is imagine I was going to select any student in a class and uh I gave you guys numbers. Let's say there are 50 students in a class. Now if I put your numbers in a piece of paper 1 2 3 4 up up till 50 and I uh shuffle the whole papers and pick one and let's say I come up with the number 14. Now I will be selecting the student with the number 14. Now that's an example of a random process. There is no predefined result.

[00:07:00]So I wasn't sure that I was going to have 14 or any other number. That's the definition of random processes. Random experiment. Now you see the word random variables. Of course that comes from number one the word from a random process and number two the second word here variables. Variables are placeholders. varying from the word varying that means you are not it's not a constant. So take for instance when you were doing algebra you always use the letter x to represent variables. So we can say x could be one in a particular expression.

[00:07:39]Uh when you put in a different uh input it could give you three. When you put in another input it could give you 10.

[00:07:47]That's the definition of variables. They are not constant values. they vary depending on whatever the situation is.

[00:07:52]So when you see the word random variables, these are obviously variables that we have in random experiment. Now let me explain to you more deeper the use of random variables in our in our when we're considering probability as a topic on its own.

[00:08:12]We had different experiments and you guys remember that we were talking about for instance we say the probability of having a head imagine we toss we tossed two coin up at the same time we say probability of having a head and a head that means the first coin gave you a head the second coin gave you a head then we went on to you know resolved it another one could say if you toss three coin up you could say what's the probability of having a head, a tail, and a head. All right, depending on whatever the experiment you're dealing with. For instance, let's use a dice. We could say probability of having a four and then would have said because I'm sure you understand that of a dice probability is 1 / 6. Right? Now all these different things you were doing here uh they are very great and beautiful and you guys have really understood how to solve them. But now take for instance we wrote probability of head head equals to uh if you toss a coin up two times of course you have four different elements in the sample space and the probability of having a head head will be one out of four I'm sure you all recolct that if you check for this you know how in this sample space of having three coin you have a sample space with a sample space of eight different elements. Now on and on and on you all understand how to really find all these probabilities. That was what we did in our last topic. Now but take for instance now that I want to go on to do some other things on this probability of head head.

[00:10:01]It's not very easy to say I want to find the expectation of this or I want to find the variance or many other things that are very useful in statistics and that's the reason why we now decide that let us assign a numerical value to all these things so that when we will be talking about probability of head now we will no longer be saying probability of head or could be saying probability of for instance x = 2.

[00:10:34]Now the aspect of statistics that gives you the privilege of expressing this as this is random variables. So now you can see what we have a probability of ed head. Now I'm writing probability of capital x equals to two. Now looking at this you want to say what exactly did I do in which these two things are the same thing. Now what I just did was I decided to assign a numerical value numerical value that means a number to this term head this outcome head. All right.

[00:11:10]Now the same way I wrote probability of head I could write probability of tail tail and in this other case now instead of writing probability of x= to 2 I could write parity of x= to zero. Now I explain to you what I'm doing. So now instead of writing parame of head I wrote of x= to 2. Instead of writing parability of tail tail I'm wring parability of x= to0. What I'm just doing exactly in this uh particular expression is I'm assigning a numerical value to the outcome of a random process. Don't forget n head ed head ed is an outcome.

[00:11:45]In this when we are having two coin we could have head ed head ed head ed head ed head ed head ed head ed head ed head ed head ed head. We could have head tail. We could have tail head. We could have tail tail. Those are four different outcomes from a random experiment of tossing two coin. I'm sure you guys agree with this. Now I could decide to assign to each of all these different outcomes when I toss two coin head tail tail head and tail tail. I can decide to assign numerical values to this so that instead of me saying probability of headed or probability of head tail or probability of tail head or probability of tail tail I can say it using a random variable expression now and how do I do that I will just say let x be a random variable representing that represents the number of x the number of when tossing two coins.

[00:12:45]Now I have defined my variable X as a random variable representing the number of when tossing two coin. This means invariably now that I can go on to say rather than writing probability of Ed now I can write probability of this random variable X now we said it's the number of so now this is two heads. So I can say x = to 2 two heads. All right. Instead of writing head tail which is just one head apply here I can write probability of x = to 1. Instead of writing tail head I can write parity of also x = to 1. And instead of writing tail tail I could write probability of x =0.

[00:13:30]All right. So I can go on now to uh use this in several other uh statistical functions. So this is just the introduction to random variables. This is definitely what random variables implies. Now you will be seeing several uh things like uh a random variable X a random variable X just understand that that use of the word random variable X is uh whenever you're going to use that word they must define to you whatever X was representing in this case X represent number of in another different question X could represent number of tails X could represent the number of lawyers X could represent the number of girls in a X represent number of boys. X represent different X represent the outcome of tossing a dice. So this is uh all the different uh there are different which we can't even exhaust because once you can think about a random experiment you can think about a random variable. So if a random experiment exists, you can have a random variable. The random variable will just be whatever numerical value you whatever you decide to assign to the outcome of that experiment. So if you are thinking about probably picking a black dog as an experiment and you have different colors of dogs in your area in that case now you can say let x be uh the number of black dogs in the community. So in that case uh when you want to say maybe two black dogs you can say probability of x= to two stuffs and stuffs like that. All right so that's just uh a brief introduction. Now you always use capital letter to represent random variables. All right it could be x it could be y and so on but the most common one is x capital letter x. All right. Now, uh I said the main one of the main reason why we make use of random variables is that it helps us to be able to uh make complex qualitative scenarios manageable.

[00:15:40]So rather than writing ed look at it now it's easily manageable this way. All right. So and uh it helps us it allows us to be able to use tools like expected value which you will probably be seeing a little bit of it in this class and also variance. We can use tools like expected value and variance to analyze the risk and outcomes of random experiment. Now while we were discussing yesterday I went on to tell you that there are two different um subtopic under random variables and that's the discrete random variable and the continuous random variable. Now I believe everybody that participated in class yesterday already understand the difference between discrete and continuous. All right.

[00:16:22]Now when you say discrete I said that is mostly that is always associated with data set that are countable. Those data set in under discrete always have integers as their result. 2 3 4 1 0 you know five six. But under under continuous we deal with measured data.

[00:16:46]When we say the height height now doesn't necessarily have to be whole numbers. Height can be 5.2 m. Height can be 3.46 m. Same thing as length as uh every other thing weights and every other thing you can think of time doesn't have to be 4 minute 4 minutes 20 seconds doesn't have to be you understand so whatever is measured we classify it under continuous data set whatever is countable whole numbers integers we classify under discrete now yesterday also I took you guys into an aspect of discrete random variable which was very very important. Now just pay attention very these things are very simple. You would not have issues with it. I said for you to be able to call something a disc random variable these two things must be fulfilled. The first of them is that probability of the random variable being equals to any value must be greater than or equals to zero. And number two the summation of all the probabilities must be equals to one.

[00:17:54]These two things must happen before you can decide to call whatever you are dealing with a ru a disc random variable. Now notice the use of the word summation here. Now let me quickly run quickly ahead then I will come back to discrete. This is for discrete.

[00:18:12]All right. For continuous data set, it's similar, very very similar. But this is the difference. For continuous also the for you to call something a continuous uh disc continuous random variable, the probability of x = to x also must be greater than or equals to z. And number two, the integration the integral of uh the whole function. Now I am using P of X but ideally I should be using F of X. I told you guys already yesterday that P of X= to X and F of X are going to be synonymous with each other under discable and continuous random variable.

[00:18:55]But it's more ideal to use f of x when dealing with integration. All right. So uh the f of this is also representing the probability. It's still this thing going on here. All right. So the same thing that happened uh in this random variable where each of each every of the probabilities of a random variable being equals to anything but like you see everything going on here now. All right.

[00:19:20]Now let's try and and put let's put a let's put a value to all this when we're dealing with this experiment. Now of course of x= to 2 when x represented the number of was 1 / 4. I'm sure you guys can understand that we said what probability is. Probability of x= to that probability of ed head is actually 1 / 4. Probability of tail tail is also 1 / 4. Probability of you having one head x = to one. Now if you're going to wrap these two things together, this will be probability of x = 1 will be 2 over 4. Of course, because you have two uh possibilities, this and this be 2 over 4. Now look at what's going on right here. This is a very precise number. This is precise. This is good.

[00:20:10]Let's try and see if we can use this to explain this. Now look at what we are saying here. But also call this of course this is a discrete random variable. It fulfills these two condition. Let me prove that to you.

[00:20:22]Look at it. Each of the probabilities you can see X being= to 2, X being= to 1, X being 0. Every one of them are all greater than or equals to Z. 1 / 4. You press on your calculator will give you 0.25. 25 2 4 1 / 2 that's 0.5 1 4 0.25 All these probabilities are all greater than zero. If it's equals to Z is still fine but it must not be less than zero.

[00:20:46]There is no negative value here. So number one condition is checked. This is good. The second condition said the summation of all these probabilities for the discrete random variable for the uh random experiment going on. The summation of all the probabilities must be equals to one. So you can go on to cross check this right here. Go on to say okay the probability of this first one was 1 / 4 1 plus the one for the second one x = 1 2 4 + 2 4 plus the one for the third one 1 / 4 1 / 4. If you press this on the calculator what does it give you? It obviously gives you one.

[00:21:21]This means this is a discrete random.

[00:21:24]This fulfills that this you can this this is all perfect. It follows the both conditions. So you get questions where they will tell you to uh prove or disprove things of that sort. Sometimes they give you questions where they would ask you to find a missing value and what you would have needed to use would have been one or of these uh uh expressions.

[00:21:47]All right. Now let's not focus so much on this. Sorry I did not complete this.

[00:21:50]The integral of all the of the function should be equals to one also. So we'll get there very soon. But this is just uh this is obviously just uh a a brief introduction. We'll be coming to we're coming to continuous random variable in a in a moment. Let's focus a little bit on discrete random variable. Now let me give you um practical examples of where you'll be needing and how questions sometimes have been framed in examination. All right. Now take a look at uh this question.

[00:22:25]Uh the question said you can write it out if you want to. A discrete random variable X has the following probability distribution.

[00:22:35]All right. And a table was given.

[00:22:41]Now the question said a discrete random variable X a discrete random variable X capital X obviously has the following probability distribution.

[00:23:01]I will explain the use of this word probability distriution probably very soon because I don't want you guys to have any problem in understanding what they are saying. So we have when uh when x was one the probability of the random variable being equals to one was 0.2 2.

[00:23:17]When x was 2, probability was 0.1.

[00:23:20]When x was 3, the probability was also 0.1. When x was 4, the probability was k. And when x was five, the probability was k also. Now the first question said you are to find the value of K and the second question was that you should find the probability that the random variable X was greater than two and the third question said you should find the probability that X was less than or equals to 4 but greater and two.

[00:24:04]So these are the three questions that uh was given uh based on the question you can see above. Now how do we solve this? Quite very simple. You need to use everything you just learned. So we have a random variable X. It has this following probability distribution. Just as I told you, it's more like the meaning of this 0.2 2. Now it's like we wrote probability of x = to 1 just like we had in the previous example and we said the answer was 0.2. Probability of x = to 2. The answer was 0.1. That's the meaning of all this. It was just given in a tabular form. All right. Now look at what we have here. You have to find the value of k. Now if you don't know what the two conditions I mentioned number one you would have a very difficult time in solving this. But if you really understand the two things I just mentioned I said number one for you to call something discrete the probabilities each of the probabilities must all be greater than zero. So far so good every one of them is greater than zero. So imagine I gave you a question and I put minus0.1. You don't need to bother solving that question because once you have minus0.1 or minus anything that can never be that can never be a discretable. The probabilities must never be less than zero. All right. Now for the second condition is what we're going to be using to find the value of k. That told us that if something is a discret variable the summation of all the probabilities must be equals to one. All right. So for us to find this number a we know that summation of all the probabilities all of them every one of them here must equals to 1. So that means 0.2 + 0.1 + 0.1 + k + k this must give me one. If this doesn't give me one then that means it's not a discret since they told me this was a discret variable this condition must be fulfilled and this is this is how we're going to be able to find the value of k. If you do this, you will have 0.2 + 0.1 + 0.1. This will give you 0.4 + 2k = to 1. 2k will be equals to 1 - 0.4. And 2k will be equals to 0.6 and k will be equals to 0.6 / 2 which will be what? 0.3.

[00:26:27]All right. So you can see exactly how to solve this. So now we have been able to use this condition to find the value of K which was given as 0.3. K is 0.3.

[00:26:40]Question number one has been solved. Now let's see the second question where find probability that the random variable X is greater than 2. The random variable X is greater than two. Now notice the word greater not equals to was used. G equals to was not placed there. So we're dealing with values that are greater than two. Now how do we solve this?

[00:27:03]Like you can see when you have this is obviously the value of your x greater than 2. x is greater and that means your x is greater than two. We're dealing with three. We're dealing with four.

[00:27:15]We're dealing with five under this range. 3 4 5 is what is covered in this range. So we have to find the probability of x = to 3. All right? or probability of x = 4 or the probability of x = to 5. And you understand the use of the word or will always give you the plus. That would be 0.1 + 0.3 + 0.3. This will give you 0.7.

[00:27:43]Can anybody give me the answer for question number C?

[00:27:47]Question number C. Find probability that X of course is less than or equals to 4.

[00:27:55]less than or equals to. You can see the word equals to is being used there.

[00:28:01]All right, less than or equals to is being used now. And uh of course for less than or equals to two, sorry, equals to four. That means we're including four. So this is among it. But don't forget we were told that x was greater than two. Greater now because I'm sure you guys can read this inequality perfectly. This means x is greater than 2 but is less than or equals to four. Greater than two means we have three. We have four but it must be less than equal to four. So three and four are the two in uh are the two uh values we are interested in. So we're going to be finding the probability the probability of x being equals to 3 or the probability of x being equals to 4. All right. And that will be 0.1 plus 0.3 that will give me 0.4.

[00:28:56]All right, I'm sure you guys are flowing quite good. This is how you tackle questions on random variables. Quite very easy.

[00:29:05]All right.

[00:29:06]Now, uh let's see another question. You guys are going to solve this on your own.

[00:29:13]uh the same format but a different uh okay I said when you see the word has the following probability distributions this is the meaning of distribution you can see how the privity are distributed so that's the meaning of probability distribution don't get it twisted here don't feel this is a big language this is what it means now we have this question where the same question but different values uh we have we have different values here.

[00:29:45]Okay. So, we have zero included in this one. So, we have the value 0 1 2 and 3.

[00:29:53]We have 0 1 2 and 3.

[00:29:56]And we have 0.3.

[00:30:00]We have K.

[00:30:02]We have 0.2.

[00:30:07]And we have 0.4.

[00:30:11]All right, the very first question is still the same. You have to find the probability of you have to find the value of K. So go on to find the value of K and give me my answer in the comment section. Then you have to find probability of X being less than two.

[00:30:28]And the last question here was for you to find the probability of the random variable X being greater than 1.1.

[00:30:38]All right. So let me see your answers.

[00:30:41]Let me see what you guys arrived at.

[00:30:44]What's the value of K, please?

[00:30:47]What's the value of K with what we just solved?

[00:30:52]What's the value for K?

[00:30:58]Can I get the answer value for K?

[00:31:02]Okay, K is 0.1.

[00:31:06]All right, let's try and do it together.

[00:31:08]Of course, we are for us to find the value of K here. We have different now you can see what's going on here. The probability of X being equals to 0 or 0.3. Probability of X being equals to 1, we didn't know that value. It was K.

[00:31:25]Probability of X being= to 2, we know it as 0.2. You can see it's more like just replicating X here. These are the values for X. Then probability of x being= to 3 is 0.4. Now so far so good. We can see that the the first rule in which we said that priority of x= to x must all be greater than or equals to zero. Tick tick tick. All of them are good except we don't of course k will have to be in check also. Now the second rule said that the summation of all my probabilities must be equals to one.

[00:31:58]This is the basis on which we're going to be able to find our value for k. So obviously we have 0.3 + k + 0.2 + 0.4 must be equals to 1. This is 0.3 to this is 59. This is 0.9 will be equals to 1.

[00:32:15]Sorry 0 k + 0.9 will be equals to 1. You can see k being equals to 1 minus 0.9 which will give me 0.1.

[00:32:26]All right. Now let's go on to find the other one. We have probity of the random variable being less than two. Now notice that there is no equals to here. There was equals to here we'll be picking two with it but there's only less than. So less than will contain the value for x being one and the value for x being zero. So that means for us to solve question number b we're interested in probability of x being equals to zero or probability of x being equals to 1. This is what will fall under the range probability of x being less than 2.

[00:33:00]Looking at what we have here, 0.3 is the probability of x= to 0 and k was that of x= 1 which we have found to be 0.1. So this is 0.1. This will give me 0.4 as my final answer. So and uh the last but not the least question we have right there is probability of x being greater than 1.1.

[00:33:22]Now when you see the word 1.1 now this is a discrete uh this question is a discrete random variable so you understand that uh 1.1 will obviously not include anything 1.2 2 1.3 this is just to confuse you guys greater than 1.1 the only value greater than 1.1 under this question is 2 and three there cannot be so don't go on to be thinking that greater than 1.1 we have 1.1.3 1.4 Before I told you guys already on that variable we only have whole numbers integers. So the values here for question number C will be provity of X being= to 2 ority of X being equals to 3 and that will be 0.2 plus 0.4 and that will be 0.6.

[00:34:13]All right. So so far so good guys. Can you give me your feedback on how you understand this topic now? Uh I feel you guys should be better now.

[00:34:25]Let me get feedbacks quickly before we move on. How do you feel now about understanding random variables? Is it making sense?

[00:34:37]Yeah, I love that comment. I said it seems easy. It's very easy. I told you already. The major point is to understand the whole once you understand all the terminologies of what is going on. You don't have issues again.

[00:34:52]That's why the beginning part is the most important. Understanding the basics is key to understanding any topic in mathematics. Now look at this question.

[00:35:02]A discrete random variable has the following probability distribution.

[00:35:18]All right. Now I'll be giving you a different format of which you can see questions so that you are not you are you're very much exposed to this also.

[00:35:32]Now the last question you had was given uh was given as a table. But in this question now you can see this is what we call a peacewise function.

[00:35:48]This is what we call a peacewise function in statistics. All right. You can see the way the the question is given now not on a table. This can be transformed to a table by the way but all about formatting. So if you see something like this, don't get it twisted and don't feel this is something bizarre or something very difficult.

[00:36:07]It's still the same thing. This is just a different way of writing. This is what we call a peacewise function. Peacewise function. All right.

[00:36:16]So you can get your questions this way.

[00:36:18]Many of many of times statistics always bring out questions using peacewise functions. All right. Now uh we're told to find the value of K.

[00:36:32]So find the value of K.

[00:36:35]And question number B said you should draw a table giving the probability distribution. So you have to draw a table given the probability distribution of X.

[00:36:54]All right. Now, uh if you can see what's going on here, the meaning of this just let me tell you guys is just saying this is saying this is the result of the probabilities. Probability of X being equals to X equals to 0.1.2K.

[00:37:10]Now this is saying 0.1 is the probability for when X was both 1 and 2.

[00:37:17]0.2 is saying this is for when X was 3 and four. and k is for when x was five.

[00:37:24]So you can go on to draw this table. I can do any of I can do this or this whichever one. If I wanted to draw this on a table, I would have written this as x. I would write 1 2 3 4 5 and then I will go on to write probability of x = to x. Now when x was 1, x was 1, the probability was 0.1. When x was 2, the probability was still 0.1 also. When x was 3, was 0.2. When x was four, pivity was still 0.2. When x was 5, probability was k. All right? So, just the same way you did this before, for you to find the value of k, now you have to sum up all the probability uh distribution and equate it to one. 0.1 + 0.1 + 0.2 + 0.2 + k must be equals to 1.

[00:38:16]This is 0.2, 0.4, 0.6. So 0.6 + k = to 1. k will be equ= to 1 - 0.6 that will be 0.4. So our value for k is 0.4.

[00:38:30]Right?

[00:38:32]So that's how you deal with uh questions under probability distribution. Now let me see how you guys this one I'm going to leave you guys to do this. I'm not going to I'm not going to tamper with I'm going to wait a little bit so that you guys can learn all you need to learn. U pay attention to the way this question is going to be written. It's also a peacewise function but in this okay let me give an extension.

[00:39:02]It's a peacewise function but having just two categories and the first category was kx the second category was k into bracket x - one this is x - one guys x -1 and this was for when x was 1 2 3 as the first one the second one was for when x was 4 and 6 all right so the same question is for you to find the value of K and also to draw a table giving the probability distribution of X. So guys, let me give you like one or two minutes to do this on your own. Let me see who gets the answer correctly.

[00:40:15]Okay. Do we have an answer yet?

[00:40:17]So we're dealing with a different uh question now. We're given a peacewise function also. We have the first as kx for when x was 1 to 3 and then k into bracket x - 1 for when x was 4 and 6. So we have to find the value of x and we're also to draw a table giving the probability distribution of x. So guys do we have an answer from anyone?

[00:40:41]Anybody with an answer?

[00:40:45]And quickly if you are in this live chat and you're yet to give us a thumbs up please do that quickly. Give us a thumbs up.

[00:40:59]Okay, somebody said K is one 14.

[00:41:13]Any other response? What's the value for K?

[00:41:27]Okay, somebody said 1 / 16. Okay, two different option answers. Two different answers there. Do we have any other answer? What's the value for K?

[00:41:42]It looks like 1 16 retracted.

[00:41:49]Okay, I'm still waiting for other people. Can we get answers? Probably.

[00:41:52]Let me give you like 60 seconds more.

[00:41:58]60 seconds, please. What's the response?

[00:42:02]What's the value for K? All right. It doesn't seem like it looks like some of you are having a hard time in solving this. Now, uh look carefully. This is a peacewise function that you have to solve. It's quite very easy.

[00:42:14]uh look at you're told that x is 1 2 and 3 and you're told in this case x was 4 and 6 how do you tackle this very simple now I would prefer I can actually solve the value of k before giving the table but just for visualization so that you guys can see it a bit uh better let me try and construct the table so that you can see it now let me write x here let me write p of x = to x here right let me give for when my x was one so we're told that when x is 1 2 and 3 the value for the probability was kx all right now that means when x was one you put x as one here that means the value of the probability will be k that's k * 1 all right now when x is 2 the value for the probability will be k * 2 that will be 2k when x was 3 the value of the probability will be 3 * k that will be 3k.

[00:43:17]Now when x was four it was a different pattern. It wasn't following this. When x was 4 that will be 4 - 1 * k. 4 - 1 is 3. 3 * k that's still 3k.

[00:43:32]3k is still the value for four. Now notice that there is no five. So don't add your five there. There is only six.

[00:43:40]So for 6 you have 6 - 1 that's 5 * k that's 5k.

[00:43:46]So this is the expression that gives you uh a summary of what you were told. Now how do you go on to uh answer this? You are asked to find the value of K. From what I've told you, the probability the summation of all the probabilities must be equals to one must be equals to one. This means invariably that uh the k + 2k + 3k + 3k + 5k all of these must be equals to one.

[00:44:25]So we have this is 1k 2 this is 3 6 9 14. So we have 14 k equals to 1. In this case you see that your k will be equals to 1 / 14. That's the value for k. All right. So you are told then to draw a table giving the privacy division of x.

[00:44:50]Now uh given the probability on x this k values are not accepted because this doesn't give you probabilities these are all still with unknown values. So you have to go on to impute the value of k into each and every one of all these things and you will get the subsequent probabilities for each of the values for x. You having this as 1 / 14.

[00:45:15]This will be 2 * 1 14. This will be 1 / 7.

[00:45:20]And the next one will be 3 over 14.

[00:45:25]This will also be 3 over 14.

[00:45:29]And this will be 5 over 14.

[00:45:34]So this is how simple it is. You just need to follow the same um process. Now uh I want to run quickly into what we call cumulative functions. Very very important. Also I want I don't want you guys to miss out on any aspect of this topic. So this is uh beautiful. I'm sure you guys uh are enjoying it so far and I can assure you you will keep enjoying it this way if you stay focused. Now let's talk about cumulative functions. Now from the word cumulative uh I'm sure you guys uh are very familiar with the word cumulative when we're dealing with uh cumulative frequency cough. I'm sure you can recollect that what we did when dealing with cumulative frequency curve was that we had frequencies we had frequencies we had data points then we had a frequency let's say data point was 0 1 2 for instance we said this let me add a three there now let's say zero will call three times let's say one occ five times let's say two called six times let's say three called one time when we're trying to construct the cumulative frequency I'm sure you guys could recollect that what we we're doing was the first one will always be the frequency and that was starting because there was no pri nothing prior to heat.

[00:46:57]All right. Once to get the next one we're going to add 3 + 5 that will be 8.

[00:47:03]8 + 6 we give the next one that will be 14 and 14 + 1 15. This was how we constructed cumulative frequencies uh under the OG curve the cumulative frequency curve. Now this gives you a peak into uh what we'll be doing when we talk about cumulative functions. So cumulative functions follows the same trend. Now look at what we have here.

[00:47:36]These are probability uh distributions for each of all these things. Now just uh look at how this works. Let me uh use this same table. Let me just edit out.

[00:47:49]Let me edit out the values because this is a bit this fractions may not give you a bet a good experience on that. Let's come from a simple standpoint then we would move on from there. So let's say we have uh the probabilities of this let's say a random variable if this random variable has the following probability distribution let's say we have 0.15 0.25 0.1 and uh 0.2 2. All right. Now, this is the first question I was asked here. You have to draw a table showing the cumulative function. Now, how do you write cumulative functions? Normally, I told you that P X = to X is also written as F of X. I told you guys. Notice the way I wrote my F small letter. When you're dealing with communive functions, it's always in capital letter.

[00:49:03]So if you are dealing with communive functions, always recive function. You just say find capital f of x. Always know that that's them asking you to find the cumulative function. Now when you see capital f of x, it's cumulative function. So you have to find function f capital letter f now of x.

[00:49:30]Now how do we do this? Of course very simple and straightforward also what we have to do we are told to construct uh a table right now uh the other questions were let me write the other questions right here. So the first question said we should find uh draw a table. The second question said we should find capital letter F now of two. And the third question said we should find capital letter F of 3.1.

[00:50:03]All right, that was the questions associated with this uh question. Now, how do we construct the quality function? First, this is the normal private division function I was given.

[00:50:16]Look at it right here. This is it. For you to construct the communive is still the same way you did that of the oif. Of course, you have your x, you have your 1 2 3 4 and 5.

[00:50:36]All right. Now, this was P of X= to X, which was still the same thing as small letter F of X. I told you guys already.

[00:50:44]Well, in this case I'm going to be writing capital f of x. Now this is the cumulative function. This is a table showing the communive function. Now just like you did in oj the first will still be the first because nothing was there was no uh nothing preceding it. All right. So we have 0.15 but the second will not be 0 will not be 0.25. Now it will be the addition of 0.25. The cumulative I'm sure you understand the use of the word cumulative. That means the addition of all the prior the previous uh ones the previous data set. So this will be 0.25 plus 0.15. This will give you 0.4 right here. For that of three, this will be 0.4. Any let me let me say it in a more better way. If you want to find the first one, repeat it. Second one had these two. Third one had this three.

[00:51:36]Fourth one had the four. Fifth one had the five of them. So for me to find the third one here, I'm going to be adding 0.1 plus 0.25 plus 0.15. And that will give me what? 0.5.

[00:51:49]For me to find the fourth one, I'll be adding the fourth one plus all the prior ones. 0.3 + 0.1 plus all these. 0.25 plus 0.15. That'll give me 0.8. And to find the last one, that would be 0.2 + 0.3 + 0.1 + 0.25 plus 0.15. That must give me the value of one. All right. So one way you would know that you are doing the right thing is that at the end of your commulative function whatever table you have drawn you must have one as the as the last end and I'm sure you understand the meaning the reason why that is because the summation of all the probabilities must always be equals to one. All right, good. Now, let's try and solve the second question. I said we find the function two. Guys, what's the answer for this? Can you give me the answer? I expect everybody to be able to answer this. What is F2?

[00:52:41]What is F2? Quickly, what is F2, guys?

[00:52:59]Do we have answers yet? What is f_sub_2?

[00:53:03]Okay. Yeah, that's very simple. F_sub_2 is the that means when x was 2. That's 0.4.

[00:53:13]You can see from this table 0.4. Now what is f3.1 guys?

[00:53:24]What is F3.1?

[00:53:30]Let me see who's going to get it. What's F3.1?

[00:53:40]Why are you writing zero? Now f3.1 I told you guys already when you see all these decimal points under discurren just understand that they are not consequential this is more like saying the cumulative function for threeative function for three so if I was using this table I just go on to pick 0.5 now if I was not using was from let's say I was speaking from here meaning of three f charact now function the point function 3.1 3.1 is above three.

[00:54:15]There is nothing there is no value for 3.1. So everything before up and before three that's 3 2 and 1 you add all these things together 0.1 plus 0.5 plus 0.15 that give you that was how we got 0.5 obviously. So that's how you get f3.1 right? So you need to understand all these things. If I say okay guys what is f4.5 what is f4.5 what is f4.5 yeah f4.5 will be 0.8 obviously everything up and including 4.5 now up and everything up to and including 4.5 the question is do you have anything like 4.2 4.3 4.4 4 4.5. No, obviously.

[00:55:06]So, the only values you have will be that of four. So, you add everything up.

[00:55:10]Okay, let me see you guys do this uh just something very similar. Uh let me see you do this on your own. Now, uh let's say we have just some value changes here.

[00:55:27]So, let's say we have a change in all these values.

[00:55:35]We have 0.2 here.

[00:55:39]We have 0.4.

[00:55:48]We have 0.1.

[00:55:50]We have 0.1 and we have 0.2. So guys uh solve the same question assocated with this just so that you guys can solve the same question draw a table showing the cumulative function f of x capture f in this case and then go on to find f_sub_2 and go on to find f3.1 so let me the same question assocated different tables but the same So let me get answers quickly.

[00:56:45]Okay, somebody said f_sub_2 is 0.6.

[00:56:48]Okay.

[00:56:50]What is f uh 3.1? That's correct.

[00:56:53]f_sub_2 is 0.6 for this question.

[00:57:47]All right. So you got your F2 to be 0.6 and your F3.1 to be 0.7. That's actually correct. So that means you guys really understand what we have been doing so far. So good. That's great. Now uh look at the question we have right in front of us. I want you guys to attempt this.

[00:58:04]You have to find the value of C. Value of C. C being an unknown value here.

[00:58:11]Find value of C in the probability mass function FX = C into bracket X - 3^ 2 where X = -2 - 1 0 1 and 2. Now guys, this brings me to something I want to quickly talk about. Uh when you deal with discrete random variable discrete we always identify with probability distribution function or we say probability mass function.

[00:58:45]Please note that these are the two you can see any of the two. Of course you have been seeing more probability distribution but you need to understand that one of the most statical way of calling this is probability mass function. So you can see it being used here. See in the property mass function the function on which your discrete discrete uh event this discrete random experiment is defined on is always what we call the property mass function. Now that will lead me to I'm sure you probably ask the question so that you probably ask a question that says why is what's that of continuous is it going to be the same thing what's the what do we call that of continuous well for continuous it's we always refer to it as probability density function probability density function all right so we say probability mass function when deal with discrete we say probability density function when dealing with continuous. All right.

[00:59:45]Sometimes you may hear the word priority distribution function also with discrete. Now but guys I need to say this uh statistics is u this is the normal way we call these things but uh not every exam may mirror this because I've seen a question before that talk about discrete random variable and still use the word privacy density function.

[01:00:06]In that case you need to understand that that was probably an examiner that was not very well versed statistically. So priority destiny function is always used for a continuous uh random variable question not discrete but if for any instance you see them using priority density function for discrete just follow true but by standard this is how we call these things. Okay so I'm seeing some questions somebody said 1 over 55 somebody else said one over 55.

[01:00:33]Okay. So you guys are saying 1 over 55 as the answer. Can anybody confirm that?

[01:00:38]And this is that's the answer for for this question.

[01:00:53]Please try and note these things I said somewhere in your notes.

[01:01:01]Okay. Many people are saying 1 over 55.

[01:01:04]It looks like we are having a consensus around 1 over 55. Let's try and solve this together.

[01:01:13]Okay. So, uh the values we have to find value of C in the property mass function. Now, it's not every time you have to use a table. All right? Or you may decide to just solve normally. So I'm going to solve without using a table in this case and u I'm sure you guys should be able to flow with it. Now this is the value for x. This is the range of value for x for this function.

[01:01:43]x is -2 - 1 0 1 and 2. Now we already know that the summation of our f of x of course in this case now our x is ranging from -2 up until 2 must be equals to 1.

[01:01:58]All right. It must be equals to one. Now this is the function. Let's put our x's into this function. Now this is the function. That means we are going to be saying summation of when x = -2 to 2. We still see x - 3² must be equals to 1. So let's put for when c when x was minus2.

[01:02:18]So we having c we have -2 - 3². All right. That's that about that plus we are using the summation adding all of them. Now that of -1. So we have c -1 - 3 square plus we're putting for 0 next plus c. We put 0 - 3 square. We are adding the next one will be for 1.

[01:02:45]All right. So we're putting for one.

[01:02:46]Next we have c 1 - 3 squared. And the next one we're putting for two. C 2 - 3 square. All this must be equals to 1.

[01:02:59]Let's try and see what was going on here.

[01:03:02]Uh, this is C - 5 2 + C - 4 2 + C - 3 2 + C - 2 + C - 1 2 everything must be equals to 1.

[01:03:25]Now if you go on to make the square of this, this will give you 25 C - 5 square is 25 + 16 C - 4 square is 16 + 9 C - 3 square is 9 + 4 C - 2 square is 4 + 1 C everything= to 1. So go on and press this on the calculator. We have 25 + 16 + 9 + 4 + 1. Let's see what the answer will give you right there. So if you add all these things together you should get 55 C = to 1 right and from what we have here you can see that our C value then we= to 1 over 55 right how you guys really understood is okay. Uh can you indicate if you really understand what we are doing?

[01:04:30]Please give me feedbacks quickly.

[01:04:34]All right. So this is how simple I told you if you are flowing with us you can't go wrong. This is the same way we're going to keep moving into every other thing and you will really really really really enjoy it. It's quite very interesting. Now uh the next part around rand this random variable which is quite very important very very important is talking about expectations and variances. Don't forget I told you that one of the reasons why we uh deal with random variable is so that we can be able to find expectations and variances.

[01:05:09]Right now how do you find expectations and variances when dealing with random variables? Now I'm still under discrete random variable. So whatever we're dealing with here classify it under random discrete random variable. When we get into continuous random variable, I need to stress that it is very important you are versatile in integration because as you saw already, we're making use of integration when dealing with continuous. All right? But when we dealing with discrete, we make use of summation. You can see the way we are adding everything in discrete. But on that continuous, we'll be making use of integration.

[01:05:49]So mathematical expectation on that discrete random variable. All right.

[01:05:56]Now I want to go straight to the point. I don't want uh any debris around this. I want you to just understand exactly what you have to do when we have uh when we're dealing with PMFs.

[01:06:10]So let's say we have X or we have random we have a random variable. We have the probability mass function F of X or probably wrote it as P of X= to X.

[01:06:22]Whichever way you decide to write it.

[01:06:24]This is the PMF. Of course, this is what we call the PMF or probability distribution function. Now, we write the expectation as E X. All right, this is how we write expectation.

[01:06:36]Now, under discrete, this is how you find the expectation of a discret variable. Guys, pay attention. It is always going to be summation of x * f of x. Now I'm sure you know we can also write it as x * p of x = to x just so that you guys don't get confused anyone you see. Now this means if you can look clearly at what's happening here uh that means when we want to find the expectation sometimes I need you to understand that sometimes this word expectation is always sometimes used as mean the mean all right the mean or the expectation you go on to multiply each of the x with their respective PMF.

[01:07:31]Let me just give you a quick uh understanding of this. So very simple you guys will ease when you see how simple these things are and this is I'm telling you. So without my word this is how questions questions do come out on what you're seeing right here. It's that simple. When you write one let's say we have 1 2 3 4 and five and we have 0.1.2 uh this is 0.3 0.4 four that's 0.7 0.1 and 0.2 I believe this should add up to one. Yeah. So let's take for instance we have this what we're saying right here and I say find E of X and I gave you this and I said find the expectation.

[01:08:13]All right. You know, the only thing I'm asking you to do is go on to do this.

[01:08:16]The summation, of course, I'm sure you know that if I wanted to write this, I I I I always love to not um bring in things that will confuse you guys. But guys, I'm sure you understand if I wrote this as I I and I wrote I is equals to one or whatever the values you're dealing with to N. It may not be one. It could be anything. But this I this this meaning of this one here is saying the first value not meaning this is not mean this I being one here is not meaning this one two no this is saying the first value of your x the first so please try and understand what I'm writing is not don't get confused at all so we have in this case we have 1 2 3 4 5 that means our n is five the first one is this one the second one is this one so this is a standard way of writing it we have to find the explanation of this what do we have to do very simple I just say 1 * 0.1 1 x * its corresponding mass function. All right. Plus, because I'm dealing with summation, 2 * 0.2.

[01:09:18]All right. Plus 3 * 0.4 + 4 * 0.1 + 5 * 0.2. Whatever this gives me is the expectation. It's that simple.

[01:09:38]There's nothing else to say around this.

[01:09:40]Just understand that once you asked to find the expectation or the mean, it's always the x values multiplied by their corresponding respective uh probability mass function. And that's the that's the whole thing you have to you have to just add everything up together and that's all. All right. Now the question will be uh for you to then ask how do we find the variance?

[01:10:09]All right, how do you find the variance?

[01:10:13]Now I'm going to talk about the variance before I then talk about some properties. Now this is where they used to hook student. That properties I want to mention is the place where they used to really catch students. All right. And I will make it as simple as possible for you. Properties of expectation properties of variances are where students sometimes miss it out on. Now let me run quickly to how to find the variance of a discrete random variable.

[01:10:39]This is for mean or expectation. All right, the variance how do we find it?

[01:10:45]The variance is going to be, look at this, e of x² - e of x all².

[01:11:02]This is how you find the variance. So guys will be asking me what exactly are we saying here?

[01:11:13]I already mentioned I already mentioned this what we have inside here. This is it. This is what you just we just talked about. So let's assume we're trying to find for this question. Of course, whatever answer you got here, let's just assume assumption. Let's say we got our mean to be 1.5 for instance.

[01:11:38]This place means that mean you can see this E of X the mean the expectation all square. So this question this if I was going to find the variance here I will talk about this very soon but let me try and talk about this first something minus if this was the mean 1.5 square that's the meaning of this all right now this one we have right here let me explain it to you what how do we find this now e of x² what how you find e of x² is that you have summation of x² f of x.

[01:12:16]So if you wanted if you wanted to find e of x², you would need to solve something else where you would if this was the question. Now let's say this was a question I was given and I wanted to find e of x² I would say 1 square time it probability for mass function 0.1 + 2² * 0.2 2 + 3 2 * 0.4 + 4 2 * 0.1 + 5 2 * 0.2. This is the way to solve uh e of x². All you need to do is just is very similar to that of e of x only that in that case your x will be squared. when you're deal with dealing with E of X square, you follow the same pattern, add everything up together and that's how you get that. So whatever answer you got here, let's assume you got let's say the answer was 2.5 here for instance or maybe I don't know just put something maybe 5.0 I'm not I don't calculate this obviously. So if you have gotten that that would have been 5.0 minus 1.5 square that's the way you're going to get the variance.

[01:13:36]All right. Now let me just recap one more time.

[01:13:40]We are told to find the expectation on that discret variable. I said the formula to use is this where you multiply the x which it with its PMF. You see how we did it here. That's how you find the expectation. This time this plus this time this plus this time this. Whatever answer you get there, that's your expectation. No long story. Just get your answer and that's all. Now for you to find the variance. The variance formula is different. This is E of X² minus E of X all square. This E of X² that was what I just explained how you're supposed to find it. You find E of X square by first of all squaring the X before you multiply with the PMF.

[01:14:21]That's the difference. Then you had it with the square of this one also with the PMF plus the square of this with the PMF and so on and so forth. That's how you find this one. Then minus whatever your expectation was, you square it.

[01:14:33]That's the way you find the variance.

[01:14:36]All right. So this is uh a good place.

[01:14:39]Now let me now go into the properties.

[01:14:44]Now this is where as I told you already where a good examiner hooks students because if you can see how very simple these things are. You'll be wondering that is it this same exam I'm going to be sitting for. Yeah, it's the same exam. And if you are good in what you're doing, that's the reason why people get their haze. It's that simple. Once you have the right understanding, these questions come out on objective very well. When they will draw a table for you and they will tell you to find the variance or find the expectation, it's it comes out. All right. The only reason why you're enjoying is because you understand it. All right. Now, let me talk about the properties under expectation under variance. This is very vital for you to pay attention to. Uh I have seen a question on that theory uh theory where they say e of find e of 5x that's where it gets complicated for students because I just explained to you how to find e of x but nobody said anything around e of 5x. All right. So you understand why it's important for you to note these next things. I want to talk about the properties of expectations and variances.

[01:15:59]Now of this is obviously of discretion variables. I'm sure you guys are very much aware of where we are on.

[01:16:08]Now uh the first property is that e of tx which tx which resembles exactly what I just wrote there is always equals to t * e of x.

[01:16:25]Now I want you I told you guys already that e of x is also uh what we used to call mean. So sometimes you can see like this mean and expectations are the same thing. I'm sure you guys understand. Uh this is the most important thing when you have the meaning of t here. T here is a constant a constant value like five there. It could be anything. It could be 10. It could be uh it could be 7. It could be 10. It could be 20. E of 20 x. The meaning of that is that if you going to find e of 5x, what you would have needed to do would have been to say 5 * e of x. So you go on to find your e of x first. Guys, can you see that? Then multiply with five.

[01:17:07]That's how you get E of 5X. All right.

[01:17:10]If they tell you to find E of 20 X, all right, what they needed you to do by this property is for you to go on to say 20 E of X. So you go on to find your U of X following the method I just explained and whatever answer you got, multiply it by 20. That's very simple, right? Good. Now let me go on to the next one where you have another scenario where you have e of t now plus x.

[01:17:41]Now we're dealing with t as a constant.

[01:17:44]How do you find this? This is another very important one to take note of.

[01:17:51]This is obviously going to be E of T plus E of X.

[01:18:02]Note all these properties very very important. As I said, E of T plus E of X. All right.

[01:18:09]Now, what is the expectation of a constant value? That means what is E of 7? Now, what is E of five? Because that's meaning of T here. E of T is a constant value. I want you to know that the expectation of a constant value is always the constant value. E of 7 is 7.

[01:18:26]E of 5 is 5. So that means in this case now E of T will be T plus E of X and we can write this as T plus if we're going to represent E of X as mu that's that's what it's going to be. So these are important properties of expectation that you need to note so that tomorrow when they ask you to find E of 7 + X you don't feel confused. All right this will be the same thing as 7 plus E of X following this rule. All right that will be that will be E of 7 plus E of X. E of 7 is 7. E of X is E of X. All right? So, whatever answer you got as your E of X, you'll be adding it with seven. All right? Does it make sense, guys? Can I get feedback? Are you really understanding?

[01:19:34]Okay. So, uh that's very important. Now let me go on to another property which is quite very very important. Imagine you are dealing with two different independent variables. I told you already. So let's say we're dealing with uh two random variables X and Y.

[01:19:57]So we are to find E of So I need to make a correction. This X here is supposed to be a capital X. All right. It's still the same thing. It's supposed to be capital, not small. Sorry for that.

[01:20:13]Now imagine we have two different uh two different uh random variables. I told you already capture data. Now what are the properties we can bring out from situations like this? We know that E of the first random variable plus the second random variable will be equals to E of X plus E of Y.

[01:20:42]All right, very very important very very important. E of X + Y that means if you have two independent variables X and Y the expectation when you are having both of them is seen as this. All right, I'm sure you guys can decode that from what we had here. Don't forget when we had E of a constant plus X, we said it's E of a constant plus E of X. So the same thing applies here.

[01:21:11]All right.

[01:21:13]And when we have a sation where we are having to do with the multiplication of the two just like we had this also we said uh you know the reason why we had this this is obviously the same thing as writing E of T * E of X but because E of T is T that's why we had it like this E of T is always going to be T. So this one also will be E of X * E of Y. All right. So note this that when you are dealing with two independent random variables, these are the two properties associated with them. Now let's run quickly to variances. What are the properties that exceed exist under uh variances?

[01:21:58]All right. So I already explained to you how to find variances and expectations.

[01:22:02]So let's see what are the properties you need to take note of. very very important guys. Experience is a very good teacher. Uh I can tell you for certain that you need to note all these things. They are very very vital for your success in this topic. Now number one property the first property here if you're dealing with the variance of a random variable plus a constant.

[01:22:37]All right.

[01:22:40]What's going to be the result? Also, if you're dealing with the variance of a constant times the random variable, what's going to be the result?

[01:22:56]Now, this obviously will just be the variance of X.

[01:23:03]The reason for that is that the variance of a constant. Now note that I said the variance sorry the expectation of a constant was the constant. In the case of variance is not like that. The variance of a constant is zero. All right. Expectation of a constant is a constant. Whereas the variance of a constant is zero. So variance of x plus var variance of x plus variance of t will be variance of x alone because variance of t is zero.

[01:23:40]Variance of a constant t is zero. Note that. So this is the result when you are dealing with variance of x plus a constant. Now we are dealing with variance of a constant time x. Of course this is going to be t² variance of x. Please note that this is how you deal with variance very important properties. So uh variance of a constant * x time a random variable x it's not variance of t you know it's not it's t² variance of x.

[01:24:30]Note this very importantly because uh tomorrow if you are told to find the variance of 5x let's say you're given a question of course when we are told to find expation of 5x we said the same thing as 5 * expectation of x don't forget that but that of variance is different variance will be 5 square time variance of x that's the way you solve for variance very important important property.

[01:25:02]Very important property.

[01:25:05]Let's go on to something very important also. In variance, we're dealing with properties. Now, these are where you will be seeing questions that uh people that don't pay attention to these things, they will get stuck. So you know how to find expectation of a random variable X variable X but you don't know how to find you don't understand the properties it sometimes will bring you uh it sometimes will uh those are one of the reasons why some students have issues in exams. So what's the variance when you have two different random variable X and Y? What's the variance of X minus Y? All right. And what's the variance of X + Y? Very important for you to note this. Uh I believe this has been seen in a question before one of these. So note it very much.

[01:25:56]Now we're dealing with variance of X and Y. X - Y don't forget X and Y are both independent variables in this case.

[01:26:08]That mean they don't depend on each other. Of course, the variance of X - Y will be variance of X plus variance of Y minus the coariance of X and Y.

[01:26:23]Now since the coariance of X and Y X and Y being independent will be zero, that means the variance of X - Y will be equals to the variance of X plus the variance of Y.

[01:26:40]All right, note that very much.

[01:26:43]Variance of x - y is not variance of x - variance of y is variance of x plus variance of y. Let's look at that of uh the variance of x + y. Variance of x + y is variance of x plus variance of y plus 2 coariance of x and y.

[01:27:07]All right. Obviously coariance of x and y is zero. So this obviously will also give me the variance of x + y to be equals to variance of x plus variance of y. Are you noticing something going on here? Variance of x - y is variance of x plus variance of y. And also variance of x + y is variance of x plus variance of y. Take note of this. These are some of the places where uh questions can come out from that could be uh quite tricky for students when they ask you to find variance of x - y. Your mind will first want to tell you that the variance of x - y is variance of xus variance of y.

[01:27:46]But that will not be the case. All right. Now let's go on now to solve vivid examples. So I want you guys to pick up your pen now and solve uh along.

[01:28:02]So how do you guys feel about uh it so far? Is there any is it difficult yet or is still very simple?

[01:28:37]All right. A random variable X with assigned probabilities is defined by Now we have x - 4 3 2 probability of 1 / 4. This one with probability of 1 /2 probability of 1 / 4.

[01:29:33]So you have to find number one the expected value of X.

[01:29:42]Number two the variance of X.

[01:29:48]Number three the variance of X + 5.

[01:29:57]And number four, the standard deviation.

[01:30:12]All right.

[01:30:14]So this is the question right in front of us.

[01:30:21]So, can you guys uh push on this now and let's see how you fair with it.

[01:30:39]So, the first question say you have to find the expected expected value of x.

[01:30:57]So uh don't forget I told you the expected value of x is always given as the summation of x times f of x. f ofx can be written as p of x = to x also which represent the probabilities. All right, good. Now, anybody here got an answer for the expected value yet?

[01:31:25]All you have to do for you to find the expected value is to go on.

[01:31:32]This is your x value associated with this probability. More like if you know you guys have to be very uh exposed to peacewise functions. This is more like they wrote X being minus4 and probability of 1 / 4 was with it. X being three probability of 1 /2 was with it and X being two probability of 1 / 4 was with it. That's just how it looks like. All right. So in this case now you'll be having x * the probability function that -4 * 1 / 4 + 3 * 1 / 2 + 2 * 1 / 4.

[01:32:22]This will just give you -1 + 3 / 2 + 1 / 2.

[01:32:32]So if you go on to solve this, you will get the value of 1.

[01:32:37]All right. Good. Can somebody tell us what the variance of x will be? How many of you got one? Okay, I can see many of you got the value one. So we want to find the variance of x.

[01:32:52]The variance of x will obviously be as I said already the expected value e of x² minus e of x all².

[01:33:07]Now luckily we have gotten our expectation of x but we have not gotten the e of x². So we have to go on to find e of x². We have gotten this e of x as 1.

[01:33:20]But we have not gotten this. So we have to go and find this this particular one.

[01:33:26]So how do we find that? I told you already e of x² is equals to what? - 4².

[01:33:43]Please note that this minus uh don't let me. This is - 4² * 1 / 4 + 3 2 * 1 / 2 + 2² * 1 / 4. This will be 16 * 1 / 4. That's the reason why I had to put it in bracket so that you don't mistake 16 for - 16. You know it is x².

[01:34:12]to the whole of x². This will be 9 * 1 / 2 and that will be 4 * 1 / 4.

[01:34:25]Looking at what's going on here, we'll be having this will be 4 because 16 / 4 is 4 + 9 / 2 + 1. So this will obviously be 5 + 9 / 2 and that will give me a final answer of 19 / 2 which I believe should be what?

[01:34:54]What should be the answer for 19 /2? Is it 8.5 or 9.5?

[01:35:16]All right. So that is that for 19 or for the variance. All right. Now let's run to variance of x + 5.

[01:35:33]the variance of x + 5.

[01:35:36]Now this is where you need to remember your properties. Variance of x + 5. Of course the property yesterday said this would be the same thing as the variance of x because I told you already.

[01:35:52]Okay. U sorry this was just for Thank you very much. This we are not done with the variance of x. The variance of X was 9.5 - 1 square that be 9.5 - 1 that will be 8.5.

[01:36:10]All right. So the variance of X is 8.5.

[01:36:15]Now question number three. The variance of X + 5. Of course I told you already that the variance of a constant is zero.

[01:36:22]So the variance of X + 5 will still be the same thing as variance of X. So this answer will still be be the same thing as the previous answer that be 8.5.

[01:36:35]All right.

[01:36:36]And what is the next one?

[01:36:42]Uh standard deviation.

[01:36:46]So guys, I'm sure you recollect what standard the standard deviation is always the square root of variance.

[01:36:55]So for you to find the standard deviation of X, you have to square root the variance of X. That will be the square root of 8.5.

[01:37:08]That should give you about 2.92.

[01:37:14]All right.

[01:37:16]So guys, you can see how interesting random variables is. Uh two more questions before the end of this class.

[01:37:23]Uh then tomorrow we will be continuing from from where we stopped that will be we'll be starting with continuous random variable.

[01:37:48]So a discrete random variable as the following probability distribution.

[01:38:00]We have X we have P of X = to X we have one as 0.2 2 as 0.4 4 3 as 0.1 4 as 0.1 5 as 0.2.

[01:38:22]So these are the questions under that you have to find expectation of x you have to find expectation of x² and you have to find the variance of x.

[01:38:38]All right let me get your answers. This is for you to solve.

[01:39:04]So can I get answers? This is for you to solve. I would really leave you to Use this as uh a consolidation of your knowledge.

[01:39:15]It's always good practice to once you understand something is good practice to practice more questions on it. That's what these two questions are for.

[01:39:30]All right. So guys, uh what's the first question?

[01:39:36]What's your expectation of X? Okay, I can see everybody writing expectation of X as 2.7.

[01:39:44]All right. What was that's question number A. What is the expectation of X²?

[01:39:55]Anybody gotten that yet?

[01:39:58]Expectation of X².

[01:40:01]Okay, I can see 9.3.

[01:40:09]Hey, do we have a concessors around 9.3?

[01:40:16]Let me wait for others to give an answer.

[01:40:21]Okay, somebody else has written 9.3.

[01:40:23]Somebody else has written 9.3. All right, so the answer for question number B is 9.3.

[01:40:30]What is the variance? That's question number C.

[01:40:34]Of course, that will be 9.3 minus 2.7 squared and that will give you 2.01.

[01:40:45]Right? Now guys, solve this one as the last question here and solve the same thing, the same question here.

[01:40:52]Expression of X, expression of X square and variance of X for this particular table. This is just to ensure that your knowledge is perfect.

[01:41:03]All right. So for this question now, what is the expectation of X?

[01:41:11]Anybody gotten an answer?

[01:41:14]Somebody wrote 1.7.

[01:41:16]Do we all agree with that?

[01:41:20]1.7. Somebody else wrote 1.7.

[01:41:25]Somebody wrote 1.7 also. All right. So we can assume and not assume but we can assert this as our answer.

[01:41:36]That's actually the answer. What about the second one? Expression of x² 3.9 3.9. Okay. The e of x² is 3.9 and uh of course the variance will be 3.9 - 1.7 squared.

[01:42:05]That should give you an answer of what?

[01:42:08]1.01.

[01:42:11]All right guys, uh this is where we have to draw the cutting. Thank you so much.

[01:42:16]Uh uh it's really great. I love the fact that you guys really understand this.

[01:42:22]That's of obviously the main reason why we are here. And uh this is just as simple as it gets. There is nothing more to this. I need to assure you guys. So uh make sure you keep whatever knowledge you have. Don't lose it. Uh if you still don't understand it when the video is out, you can obviously go back to it. U find time to solve questions out there. Look for more questions to solve. Tomorrow we'll be continuing from here. We'll be having continuous random variable. Guys, uh for the uh next class, please if you are having issue with integration, please try and revise integration before tomorrow's class because we'll be doing more of integration on continuous random variable.

[01:43:15]Thank you guys. so much. Wish you all the best. Enjoy the night.