 Dear students, let me present to you an example of the computation of the variance of a conditional distribution in the case of a joint PDF of two random variables. The joint PDF that I am considering is f of x, y is equal to 2 where 0 is less than x less than y and 0 less than y less than 1. So, how will we proceed? The very first thing to note is that this rather complicated support or domain of this bivariate density function which is written in two parts which I just read out 0 less than x less than y comma 0 less than y less than 1. We can write 0 less than x less than y less than 1 or split into two portions but not the same as what it was before. We can also write it as x less than y less than 1 and 0 less than x less than 1. If you draw the graph of this domain you will find that it is easier for you to understand all these points that I have just made. Alright, how do we proceed to find the variance of the conditional distribution? Well, first and foremost, we need to find the marginal distributions. So, first of all to find the marginals, how do we proceed? Baki to simple hai jahse hota hai, sirf yey aad rakhain ke wohi because of that complicated domain, what limits you have to apply when integrating with respect to x and what limits when integrating with respect to y? Well, when you are wanting to find the marginal distribution of y, you will be integrating with respect to x and the limits have to be my dear students in this particular case, 0 to y. Jo expression aapke saamne abhi maine rakhha hai, ek expression, ek wahid jo ke us situation ko depict kar sakta hai. Usi teraf bhi agar aap dekhlein to aapko nazar aayega ke as far as integration with respect to x is concerned, the limits have to be 0 to y. Ishi tara, in order to find the marginal distribution of x, you will be integrating with respect to y and this time the limits have to be x to 1. Alright, let us then proceed. To find the marginal distribution of x, you take the integral and you do the steps and what is the result? As you can see on the screen, f of x is equal to 2 into 1 minus x such that x goes from 0 to 1. Jo marginal distribution ki domain ya support hai. In this particular problem for both distributions it is going to be 0 to 1. Again, just have a look at the domain of the bivariate function carefully and also draw the graph and then it will be clear to you that what I am saying is correct. Ishi tara, find the marginal distribution of y, do the steps, you can see them on the screen now and what do we get? We get f of y is equal to 2 y such that 0 is less than y and y is less than 1, yani y lies between 0 and 1. Alright, now that we have found the marginal distributions, next step is to find the conditional distributions. Now, what is the formula for the conditional distributions? Well, f of x given y is equal to f of x y divided by the marginal distribution of y and f of y given x is f of x y divided by the marginal distribution of x. So, just put the values, what do you get? You get f of x given y is equal to 1 over y and f of y given x is equal to 1 over 1 minus x. Ache ye to expression to aage. Lekin ab again be very, very careful in writing the domains of each of these two conditional distributions. Domain kehle ya support kehle, it's the same thing. Here, because of that complicated domain of the bivariate function, what is the appropriate domain, what do I have to do? What do I have to write for f of x given y? Students, it has to be 0 less than x less than y. And for the other one, f of y given x, my dear students, it has to be x less than y less than 1. Yani bohi jo apne integral lete vak jistra kia thena, ushi tara se you will have over here. All right, iske baad what do we do next? Next, we would like to find the conditional mean of x given y. Kyuke hain pehle ye nikalenge uske baad hain baad me kuch hot nikalenge jisse variance jo hain hain our ultimate goal hain uski tara faan jaa sakenge. So, the expected value of x given the random variable y equal to the small y, some particular value, what is the formula to find this? Expected value of x given y. Well, the formula is integral and because this integral is to be with respect to x, so the limits are going to be 0 to y. So, the integral 0 to y of x into f of x given y with respect to x. Jesa ke aapko maloom hain, hain bonyadi torpe expected value of x nikalenge hain. Isli hain me integrate bhi with respect to xi karna hain. Sirf jo baad dekhne ki hain wo ye hain, ki agar unconditional mean ho, aap simply e of x nikalenge ho, to aapka mola kya hota hain. Integral x into f of x dx. Lekin yaha because it is conditional mean, to aapko integral x into f of x given y aur aage bohi dx. Yani f of x unconditional ya marginal ke li jes ke isme of course that's the marginal one. Uski jaga pe you will be writing the pdf of the conditional distribution. Yaani f of x given y. All right, let us substitute hamara jo f of x given y abhi tori dekh pehle hamne nikala tha, what was it? 1 over y. So, hamara expression kya ban gaya integral from 0 to y, x into 1 over y dx. Isko aap solve ke li je. And what is the final result? It is y over 2. Y over 2 is the final result aur isski saath phir aapne y ki all possible values likhni hain. Yaani aap likhenge 0 less than y less than 1. Ab aap thoda sa shahid yaha pe confused ho raheon ke aapne expected value of x nikali hain. Conditional beshak wo hain. But it is the expected value of x to expected value of x to ek constant ho na chahiye. Toh students issko kishtha interpret karna hain. Isko bade gaur se suniye. Actually issko aap ishtha interpret kar sakta hain ke wo jo small y, some particular value of y, wo value that can be any number between 0 and 1. Toh for example agar main y ki value 0.5 ya 1 by 2 leonu. Toh abhi jo mera e of x given y aaya hain wo y by 2 hain toh y ki value jo 1 by 2 rakhungi toh kya aajhaya. I will have 1 by 2 or divided by 2 so that is 1 by 4. So wo constant aagya na ji phir suniye. Y ki value 0 se 1 ke bhi chma kuch bhi ho saktiye agar y ki value maine 1 by 2 lee. Toh phir meri yeh jo conditional mean hain uski value aagai 1 by 4. Toh issi tara you can have other values of y and therefore other values of the conditional mean. Iske baad issi tara students aap expected value of x square given y nikal liye. Bilkul wo hi tari ke kar aur isska result as you can see on the screen it comes out to be y square over 3 and again we have to write with it that y lies between 0 and 1. Aap jab ke yeh dono nikal aayin hain now we can very quickly find the conditional variance. Variance ka shortcut formula aam halat me kya hotta hai. Variance of x is equal to expected value of x square minus expected value of x whole square. Yeh aap aap conditional kar rahe formula waise hi ho ka. Variance of x given small y is equal to expected value of x square given small y minus expected value of x given small y whole square. Yeh ni formulae ka patent bilkul wo hi hai. Sirf itni si baad hai ke iss wak tham conditional cheez aayin aapke andar rakh rahe hain. And so substitute ka liye y square over 3 minus y by 2 whole square. Solve ka liye. The final result comes out to be y square over 12. Iske saad bhi zorur isske forend baad comma daale aur likhain 0 less than y less than 1. So this is the conditional variance of x given y. Aap bilkul issi tara you can proceed to find the conditional variance of y given x.