 So, in the last lecture, we started talking about random sampling. We discussed about what we mean by branding sampling from a particular population. We said that for us, population is nothing but some underlying probability density function and when we samples from this population in an IID fashion, we call that as random sampling, ok. And then we started talking about sample mean, sample mean and sample variance. So, these were the definitions of sample mean, sample variance and sample deviation. Then we looked into some of their properties like expected value of sample mean is nothing but the mean of the underlying population and the variance of the sample mean can be calculated to be sigma square by n. And we also said that expected value of standard sample variance is nothing but the variance of the underlying population. And we actually showed this, I just give the computation and I asked you to verify this, I hope you verified. See, notice that like if I say some computation to verify, please verify, this may come in your quiz or any place and at that time you should not repent, ok. And then we said that when the expected value of sample mean is equals to the mean of the underlying population, then the sample mean which is basically one of the statistics we consider, we call that statistics as unbiased. And similarly the another statistics we have is statistic we have is sample variance and we said that sample variance is also unbiased, ok. Now, let us move on. When you do sampling and all, we have to do all the analysis with only some finite number of samples, right. Maybe when you do samples, when you collect samples from people, maybe you will collect maybe 100 people, 200 people, 300, 400, maybe 1000 or maybe at max few lakhs. And based on those samples only you have to do analysis or try to understand what is the underlying mean and variance of the population. And always collecting samples is a expensive task. You should not assume that, ok, like I will just go and get as many samples I want, no, right. Like if you have noticed like when all these drugs are being tested, right, difference vaccines are tested, they were trying to give it to different people and see what is the effect of the drug. And based on that they were trying to come up with how efficient their drug is. So, giving it to someone and then seeing that how drug affects him that is like collecting one sample, right. Naturally asking more people to come for this drug testing is a very expensive thing. Like if by mistake something happens to anybody, that is a lot of liability is there. So, getting more samples is not always easy and you should not be assuming that, ok, samples are too like I will get as many samples. And another thing, if you want to do a go and do market survey, like if you are interested, if you are launched a new product and want to see how many people will be interested, you have to set up a whole process of getting the sample, you have to put somebody to go and talk to people. Like maybe you have to employ somebody, you have to pay him for that and he has to go and talk to people. So, his travel needs to be arranged, all these things, right, these are all, they cost money and I mean all these people who do poll prediction, right. Like you see that they put a mammoth system for this, they put lot of people, they will send to different regions, they will talk to people, that is again involves lot of money. And now things are simplified, most of these samplings, maybe people put it on just opinion polls, like just to make a some web page and keep on sharing it and maybe like a through Google form or something, ask people to give their inputs. That is like asking somebody is like you are basically getting one sample from that person. But that itself is also going to take maybe cost you something and it is not that everybody, you can approach anybody, like if you start sending everybody, everybody will just flag it as spam and they may not get any response, ok. So, lot of issues are there and that is why we should always be ready to see what kind of information we have from a given number of samples. Now, the question is how to go about analyzing this. So, to simplify things what we will do is, we will assume underlying population is Gaussian. This is our first simplification and try to understand how the analysis goes about, ok. And then we will consider what happens if it is non-Gaussian. If suddenly when the things are non-Gaussian, things becomes complicated, ok. Somehow Gaussian distribution is something which is very amenable for analysis, mathematical analysis that is why most of the time we consider. From the theoretical point of view, we assume Gaussian distribution, ok. That is why this is one of the popular distribution because it is easy to analyze. Now, when we have samples, we are going to do some statistics on them. It could be mainly sample mean or sample variance and now try to understand some of the properties of this when the underlying population is Gaussian distribution, ok. Now, if your samples x 1, x 2 are all random samples and the underlying population is Gaussian with mean mu and variance sigma square. The sample mean itself is going to be Gaussian distributed with mean mu and variance sigma square by n, ok. Let us now, is this obvious? So, what we are saying? Let us say we have this x 1, x 2 up to x n and each of this xi is mu sigma square and now we are saying x bar is 1 by n summation xi. How do you check that this x bar is Gaussian distributed? What is the method you have? One possibility is moment generating function, ok. See that what is the moment generating function of x bar, ok. How you are going to find out? It is going to be 1 upon e to the power t x, you do it for some x and you know that this is nothing but e to the power t summation xi by n. Now, I can write it as product of 1 to n expectation of e to the power t by n xi. Can I do this? Why this product is correct? Because they are independent. Now, what is its value? You know what is the moment generating function of a Gaussian distribution, right. Compute at what point? t by n here. What is that value? Can somebody tell me? Exponential plus, is this correct? I do not know, I am just taking you and now let us simplify this. Now, with this it is going to be exponential, I am adding now all of them. So, mu t it becomes mu t plus half sigma square t square by n, ok. Now, what is this corresponds to? What is distribution corresponds to? Now, if you compare it with the template of the normal distribution, this is going to have a mean of mu and variance of sigma square by n, right and that is why this is going to be Gaussian with parameter mu and variance sigma square by n. Now, x square we had. Now, what is our s square? s square is 1 by n summation. Now, what we are saying is this x bar and s square they are independent. So, on a first look does it make sense? Because they all this x bar, now we are talking about this x bar and this s square both s bar x bar and s square they all depend on the same set of samples x i. See x i is let us take one particular x i that x i is there and that x i is also there in then some. So, both of them do you expect it to be independent or at least what your intuition says like. It is hard to believe that right this x bar and s square which both of them depends on the same set of samples they are independent, ok. But for us we will go with the definition of independence. What is the definition of independence? If you compute the distribution of x bar and compute the distribution of x square and if you look into the joint distribution they should split into the marginals and it actually happens for this whenever it is Gaussian and that is why we are going to call I mean that is why it is a result that they are independent. And in fact, you can show this and this requires some good amount of calculations that is why I am skipping that and it is there in the book which I am going to share and I will refer you which chapter and which section you have to look into for this calculations, ok. What we are again going to do is you are going to find now you have already have x bar distribution you already have. Now, you need to figure out s square distribution, but finding the distribution of s square is going to be hard. Instead what you are going to do is you will look into their joint moment generating function and you will show that their joint moment generating function actually splits through that you are going to show they are independent, ok. And this is one of the important properties that we are going to use later and thanks to the special structure of Gaussian distribution this property holds otherwise if you replace this by any other distribution this need not hold another property. Now, this sample stand sample variance if you multiply by n minus 1 and divided by sigma square this has a chi square distribution with n square degrees of freedom, n minus 1 degrees of freedom, ok. So, notice that x bar has Gaussian distribution and when you s square when you divide and multiply accordingly it will have chi square distribution with n minus degrees of freedom. And by our definition we know that chi square distribution is related to the gamma distribution. So, chi square distribution with n minus degrees of freedom is nothing, but gamma distribution with parameters n minus 1 by 2 and half, ok. So, now at this point you may be wondering x bar, x bar consists like when I defined x bar it has n random variables in it and each one of them is like a Gaussian random variable in this case and s square also consists of this n random variables then why it is not n degrees of freedom like I can think of these are like n components, right and which all of them are like independent they can vary in an arbitrary fashion why is that they are not n degrees of freedom why it is that n minus 1, yeah. So, even though we have n components, but there is an x bar here, right and that x bar is affecting my s square their average is affecting and because of that you can just work out that like even with that if you take n minus 1 components the n components get fixed because of this x bar part here. So, because of that 1 degrees of freedom get reduced and you will end up with n minus 1 degrees of freedom, ok. Now, let us continue to analyze this random sampling of this Gaussian distribution itself and when we try to analyze this say I will tell you at this point what is our ultimate goal? We want to get the parameter from samples let us say we already know it is a Gaussian already we have put the structure that is mu and sigma square, but I may not know mu and sigma square mu and sigma square unknown and what I have is I have access to the data I have samples x 1, x 2 up to x n and I need to mu and sigma square from data that is x 1, x 2 this random sample. From this random sample at best I can find the sample mean and sample variance, but this x bar is it same as mu that I want mu, but this x bar is not same as mu, right. So, that mu there will be some difference this need not be 0 and similarly this s square minus sigma square this need not be same. Now, what I want to understand how much is this actual difference is? If I have to compute this only n sample I know that this is not going to be 0 it is going to be some positive value, but how much is that and how can I quantify that? Similarly this also I know this is not going to be 0, but how much is that error and I need to quantify that. So, for that now we will start thinking about how to do that and for that we will use all the Gaussian properties, because in general we cannot compute it for any distribution easily, but we focus on Gaussian distribution and try to get it, ok. So, in that process we are going to now in doing that we will end up studying something called student t distribution.