 So, we just talked about population and random sampling, sample means sample variance standard deviations, sample standard deviation, sampling we are now currently focusing on sampling from normal distribution and we just talked about student view distribution ok. Now we are going to talk about some more distribution today called f distribution and we will talk little bit about convergence of random variables. Is the convergence of random variables done in IE 6 to 1? Convergence in probability, convergence in distribution, convergence in expectation nothing of that sort ok. Let us see how much we can do that today ok. And now let us say we have actually two population denoted as normal with mean mu and variance sigma x square, another one normal with mean mu y and variance sigma y square. These are two different populations. From one population I have these samples IID and another population I have these random samples. Now suppose let us say I am interested in the ratio of their sample variances. So, what does this capture? I want to basically look into the variability of the populations that is how does variance of one population compare with that of the other. And since I do not know the variance, I would replace that by their sample variances and then look into the ratios ok. Now let us see how to compute a distribution and I will ask you ok find out the distribution of this ratio of the sample variances. How to go about it? Now you want to again appeal to the Gaussian distribution properties. Let us say how to do that? This xx square xy square I will divide it by this quantity the actual variances and if I simplify this I will get this quantity and the numerator I am multiplying and dividing by n minus 1 denominator I am multiplying and dividing by m minus 1. Well by the way notice that the population this first population I have n samples and the second population I have m samples n and m are not the same. Now if you focus on the numerator here this we know it to be chi square distribution with n minus 1 degrees of freedom right that is what we said. And the denominator is again a chi square distribution with m minus 1 degrees of freedom and this quantity we are going to call this the distribution of this we are going to call it as f distribution with n minus 1 and m minus 1 degrees of freedom or alternatively the ratio of chi square distribution with n minus degrees of freedom and m minus 1 degrees of freedom we are going to call it as f distribution with n minus 1 and m minus 1 degrees of freedom. So, we have another distribution here again this distribution is kind of popular is by this Ronald Fisher who is another famous statistician and f distribution comes from his first letter f ok. So, fine what we are basically going through is different statistics we are interested in right sample means sample variance they themselves are random variables and now we have trying to basically look into what kind of distributions they will have and that has led us to find out this t distribution and f distributions and again you can go back and compute the actual pdf this f distribution and you will end up with this pdf function which is pdf with p and q degrees of freedom and we are going to denote an f distribution with p and q degrees of freedom like this ok. Now the question is how this pdf comes again you can go back to your classical method of finding the joint distributions from the known distributions ok. Now if you notice f distribution is nothing, but ratio of 2 chi square distribution right is the this chi square distribution in the numerator and the chi square distribution the denominator are they independent of each other why they are coming from different populations which are assumed to be independent because the numerator is coming from this population and the denominator is coming from this terminal they are independent ok. So now what I will do is I have now represented this fpq as nothing, but u by p v by q where u is chi square distributed with p degrees of freedom and v is chi square distributed with q degrees of freedom and I have appropriately multiplied by p and q. So am I right in saying that f distribution and p and q degrees of freedom can be written like this if that is the case and you know that u and v are independent then things are easy for us. We can again go back and appeal to your computation of joint distribution functions of joint distributions ok. So now to find out what we will do is again I am going to define one random variable like this x which is of my interest and I am going to denote another random variable whatever of my interest I am going to call it as x and I have denoted another random variable v and r x and y are independent y x and y has to independent because this depends on u v x also depends on v. So, x and y need not be independent ok, but we know that u and v are independent ok and we know that is why joint distribution of u and v is nothing, but product of each one of them and we know do we know the CDF of u, u is what chi square distribution right we know the chi square distribution CDF right. So, we can write the distribution of joint distribution of u and v. Now that is it if you know this distribution now you can find out the joint distribution of x and y is not it. Again what you have to do is use your standard Jacobian method ok and then find the marginal of x and I will again leave it you to complete all these standard steps ok that is again nothing you have to do try yourself works out better otherwise just refer to the books that I will post it in which all the details calculations are given, but at least try yourself if you are able to reproduce that result ok. Now see like we have enlarge our scope of many distributions right it is just like now going from Bernoulli, binomial, Poisson, Gaussian, exponential. Now we talked about beta distributions, gamma distributions then we talked about t distributions and f distributions ok, but as you see gamma and beta distributions they are somehow related to my basic distributions right. What was the relation between gamma distribution and exponential distribution was there any relation ok. Gamma distribution was nothing but a summation of n independent exponential distributions with parameter lambda. So, gamma n lambda was like that and what was the relation between beta distribution and other distributions 1 1 is any other than that anything that is right beta 1 1 is ah ratio of gamma distributions beta did you say that. Now this beta distribution is the ratio of the gamma distribution can you tell like if I say beta distribution with parameter a and b. Now can you express in terms of the gamma distribution ok let us say my x is beta a b. Now you want to write x as x 1 by x 2 x 1 plus x 1 by x 2. Now what is x 1 gamma what parameters? What is n? You have to now tell me in terms of a and b right what is lambda now? You have to tell me everything now you are trying to write express beta a b in terms of gamma distribution only thing you have is a b tell me that. Now ok n what is n? So, gamma has to be n and lambda right ok. So, we said gamma is now ah alpha and lambda we said ok tell me how is alpha related to a and b alpha equals to ok ok let us take this is ok x 1 alpha equals to a and what was lambda? Yeah, but what is it? Any lambda how can it be any lambda? It has to be dependent on a and b right. I am not sure you check this your claim is if I give you 2 gamma distributions you are able to get a beta distribution by writing those gamma distribution in this format. Either if I tell you ok let us say let us say I will tell you x 1 is gamma with parameter let us say alpha 1 and lambda 1 and I will say x 2 to be gamma alpha 2 lambda 2 and now your claim is x is beta distributed with a b now tell me what is a and what is b ok lambda 1 lambda 2 is equal to what that has to be that is fine and ok you want to set lambda 1 lambda to be the same what about alpha 1 and alpha 2 and lambda 1 lambda 2 no you cannot be just greater greater than 0 means which be like I can take 10 20 30 40 like you said a equals to alpha 1 you fixed it I cannot take anything now when you say b equals to this you fixed it now fix me lambda 1 and lambda 2 b equals to alpha 1 plus alpha 2 ok lambda cannot be then like let us say I take lambda 1 equals to lambda 2 equals to 10 then this will give me some value and if I take lambda 1 equals to lambda 2 equals to 20 it cannot be same right it will give me something else ok just throw up or you know what you had then tell me then what is its value any constant will do ok I do not know about any constant ok then it is a exercise for all of you ok if you do not get you catch these two people who are making this claim ok their claim is if you take two distributions comma 1 and comma 2 with parameters alpha 1 lambda 1 alpha 2 lambda 2 where alpha 1 is same as alpha 2 and any value then you will get a beta distribution when you express like this x is like a beta distribution where a equals to alpha 1 b equals to alpha 1 plus alpha 2 that is their claim ok so I may ask you to prove or disprove this so verify this ok now let us see some simple properties of this f distribution suppose you have x to be given to be f distribution with degrees of freedom p and q and you may be interested in taking the reciprocal of that 1 by x then my claim is it will have a f distribution with degrees of freedom q and p so p and q has become q and p now why is that that that observation is obvious right because let us say you know that x can be written as u by p v by q this should be have been q where u is chi square distribution p degrees of freedom and u is chi square distribution with q degrees of freedom now what is 1 by x in this case 1 by x case now just right now the numerator is chi square distribution q degrees of freedom and the denominator is chi square distribution with p degrees of freedom then by definition this should be f q p ok just again verify ok now suppose what is the connection between student t distribution and f distribution ok now suppose x is student t distribution with p degrees of freedom then it so happens that the square of that is f distribution with the 1 and with parameters 1 and p. So, why is that again you can verify this pretty straightforward let us take x to be student t distribution and if it is a student t distribution I know that that could be represented as a ratio of two random variables u and v right actually u and square root of v by p where u is normally distributed and v is chi square distribution with p degrees of freedom that we have discussed and we also said they are independent this u and v are independent now if you just take the square of this you just take the square now this is u square divided by v by p now we have already discussed that if u is normal then what is the distribution of u square chi square with what degrees of freedom 1 degree of freedom right that is what we have discussed sometime back this is going to be 1 degrees of freedom. Now what I have done now if you carefully look into this now I have this like chi square numerator I can divide it by 1 nothing changes and the denominator is now this v is chi square distribution with p degrees of freedom right. Now notice that now I am able to write this as ratio of 2 chi square distribution where the numerator is of 1 degrees of freedom and denominator is of p degrees of freedom and appropriately normalized and by definition what is this f 1p because the numerator has 1 degrees of freedom denominator has p degrees of freedom. So, you people are talking about this third example ok let us look into that suppose let us say x is f distribution with parameters p and q if I write like this x divided by 1 by x, but x is being multiplied by 1 plus q x ok. So, what is this actually can I write it in this form ok fine let us let us take it in this form and now we are saying that this is nothing, but beta distribution with parameters p by 2 and q by 2 ok. Now let us try to solve your problem can we use this result suppose this is true you are talking about this x right x is ok you want to say x is what ok this is my right this is fpq q right. Now how is this I have to ratio take like this how this becomes beta and this is not even gamma it I do not see gamma coming gamma is not directly related to fpq chi square related to gamma that is right now try to fill in the gap like how now can you prove this using this ok. So, here this is my a and this is b according to you it is not clear what you are saying it does not come ok. Anyway you this requires some computation this is not see the claim 1 and 2 are straightforward we can just apply the definition, but this requires little more thinking ok how to derive this relation between f distributions and beta distribution. So, work out this as an exercise ok fine. So, and then try to see that if you can connect with this result if and what if it is indeed correct ok let us stop here today.