 any questions so far what you have discussed? So, we have been discussing I think in the last class we talked about the limit theorems right law of large numbers and central limit theorem. Now, as we move from probability to statistics now what we will be doing is we will be trying to look mainly at the data and trying to see that from which underlying distribution or what are the parameter associated with the distribution that is likely generating this data. Whereas, the probability was about we already took some distribution and try to understand its properties and in probability like we already start with some distributions all the parameters everything is known and according to that we talk about data being generated. Now, in the statistics we do not start with distribution directly we may start with some model of the distribution, but we do not know its complete characterization we do not know what are its parameter, but we will start we will have access to the data and from that data we will try to see that what is the underlying distribution or what are the parameter of that underlying distributions. For that let us have a quick recall on all the distributions we talked so far we will expand this in this class today, but as a recap in the discrete case we talked about Bernoulli, binomial, geometric and Poisson and notice that all of them are coming with certain parameters ok. And also in the continuous case we talked about uniform exponential Gaussian Rayleigh and I think we made some remark about other distributions like a Laplace distribution and couple of other distributions. Now, if I tell that ok data is normal with mean and sigma square I may say that data is coming according to the law which is Gaussian with parameter sigma and sigma mu and sigma square. I may just say that ok data is coming from some Gaussian distribution with some parameter mu sigma square. If I specify what is the value of mu let us say if I tell mu is 0.5 or some value like sigma square is 1 then I have completely specified the distribution and here you do not you somebody do not need to give you data you can yourself generate the data as using this parameter. So, this is like if I tell this parameter this is completely specifying your probability. This is like a probability on the other hand I may say that ok data is coming from some distribution which I am going to assuming to be mu square, but mu is unknown and may be sometimes even sigma square is also unknown. What here what I am just assuming data is going to be I am instead of assuming some arbitrary distribution I am going to restricting to Gaussian distribution, but here I am I just do not know what is that parameter mu and sigma square are. Now in statistics we will try to infer this parameter mu and sigma square from data ok. So, sometime infer we call sometime we called estimate and all like we will make that more formal ok. So, to infer this parameters we need data and that is why even though we start with some underlying probability model when I talk about statistics I will always say that we will start with data and from that data we have to find out the parameter of the distributions. Ok now as I said in we want to little bit expand our family of distributions we know and see what are the parameters they are associated with ok. The first distributions we are going to look into is gamma distribution here I cannot write so, I cannot write in this I will just use the mouse. Gamma distribution is parameterized by two parameters alpha and lambda both alpha and lambda are positive and last time we discussed the definition of gamma function right. So, we all know the classic gamma function, but that gamma function can be further parameterized ok with some parameter like if you just if you ignore this denominator and let lambda equals to 1. So, that is x to the power alpha minus 1 e to the power minus x when you integrate it between 0 to certain number that will define the gamma function at that number, but now you can bring in this another parameter lambda. So, then if you add alpha to the power lambda x alpha minus 1 and e to the power minus lambda that is like a generalization of your gamma function and now if you see that if you normalize this integral by this actual integrand you will see that this actually when you integrate it over the whole range from 0 to infinity this normalizes to 1 this integrates to 1 that is what we discussed last time right. So, this is a valid probability density function. So, this is positive valued and area under this curve is 1 ok. Now, as you see I have plotted it in two scenario in the first scenario I have hold this lambda to be constant to 0.2 and then I am varying alpha. As you see that as I go from alpha equals to 1 to 2 to 3 then alpha equals to 1 this is like a monotonically decreasing then I made alpha equals to 2 it started first increasing and then decreasing and as I am increasing alpha it looks like it is started only showing increasing behavior. So, that is why when you have when you if you look into different value of alpha it is shape is changing it from monotonically decreasing to all the way monotonically increasing. So, the shape is changing that is what we call this alpha is shape parameter and here I have plotted for a fixed alpha different value of lambda when it is lambda equals to 0.1 it was like monotonically increasing, but it also showing some small tendency of decreasing when lambda equals to 0.2 this is also increasing and decreasing lambda equals to 0.3 increasing and decreasing. So, as we are increasing lambda it is taking larger and larger values in some particular peak regions right. So, basically what it is happening lambda is in kind of scaling and that is also kind of evident from this maybe not so evident, but anyway as you see like as we are increasing lambda it is kind of scaling like y values are getting more scaled not across entire range, but in some particular regions where the peak is happening ok. Because of this sometimes this is also called scale parameter. Now, even though we have discussed this gamma function, gamma distribution in terms of gamma function it has relation to some previous distribution that we know. What is that? Suppose you take some integer n and take n number of Gaussian distributions that are sorry n number of exponential distributions coming from parameter lambda with lambda and they are all IID if you add them the resulting distribution is exactly going to follow this PDF that is it is going to be gamma with parameter n and lambda ok. So, this gamma distribution can be interpreted as gamma distribution when n is integer it can be interpreted as sum of n IID exponential random variables with parameter lambda. So, basically the gamma distributions we can now take any value of n and lambda right because these are n and lambda parameters it is not necessary we have say we have never said that so far in the gamma distribution n has to be integer only as a special case we showed that when n is integer then gamma is related to the sum of IID exponential random variables. So, now let us look into what happens if I restrict myself to some specific values of alpha and lambda. So, here I have taken alpha equals to half and lambda equals to half. So, here the first parameter not integer I cannot interpret it as a sum of half exponential random variable with parameter half I cannot interpret. But this distribution gamma half and half this has a special name and special significance this is called chi square distribution with 1 degrees of freedom and it is denoted as this symbol chi with subscript 1, 1 is denoting here 1 degrees of freedom and chi squared the 2 is the superscript 2 is denoting chi squared ok. In the gamma distribution if you are going to set alpha equals to 1 and lambda equals to half this is the distribution you will end up that is simplified one for the specific value of alpha and now you may be wondering ok fine you have taken alpha equals to half then where is this 1 degrees of freedom why that term is coming ok. And just to see that why could that why this has been given the name of 1 degrees of freedom. So, suppose let us take uniform distribution sorry let us say u is some random variable which is normal distributed. Now you can show that if you take the square of that that has this gamma distribution that is square of a normal distribution is chi square distribution with 1 degrees of freedom ok. You see that now u is I have just taken u to be 1 random variable it can take any value as per that probability density and if I just square it I am getting gamma distribution and here u is one component which can take any value it likes as per that ok. And if I am just squaring that I am getting this gamma distribution ok ok next there is another if you now this instead of a gamma half and half if you take gamma n by 2 and half ok. Now this is also given a special name this is just like extension of this chi square distribution with n degrees of freedom and here n is like an integer and that is denoted as chi square with a subscript of n. So, then what is chi square distribution with n degrees of freedom this is nothing, but a gamma distribution with parameter n by 2 and half and if I plug in the value n by 2 and half in this pdf we will we will get back this value. Now again this chi square distribution with n degrees of freedom this is again related to other distributions. So, let us say if you have n random variables n iid random variables I am now considering two cases case one where all of the random variables have normal distribution if you square them and add that will have chi square distribution with n degrees of freedom. So, when n equals to 1 when this n equals to 1 we had already this case that we have got. So, it is not just that only Gaussian is related to this gamma distributions you can even verify that if all this u is are exponentially distributed with parameter half and if you are going to add them and all of them are iid then also you will get gamma distribution with parameter n comma half notice that here it is not n by 2 it is n here and now even though I have stated them here you should check this check this this is indeed correct ok. Make sure that if you are going to add all this if you generate n iid random variables which are normal and if you are going to add them you will indeed get this chi square distributions ok. Now let me say I think at some point we should actually start simulating this and say.