 So, now some more definitions. Now, you have introduced this intervals interval estimators. So, we can define some terms related to this how good that interval is ok. Suppose, if you have an interval estimator given by this lower function L of x and upper function by U of x for a particular parameter theta, its coverage probability is the probability that it covers the true parameter ok. You understand this? So, basically mathematically we are saying what is the probability that the true parameter lies in the center well ok. So, this is what we are going to call it as coverage probability with what probability you are I mean you are covering. Now similarly, if I have an interval estimator given by L and U functions for a particular parameter 3, its confidence coefficient now is the smallest value of its coverage probability across the parameter set. So, we just said this is the coverage probability and my theta can come any value over my parameter space right. That is something which is not in my control theta can be any value where theta like now I have to look at the worst case. So, I will take the smallest value of this coverage probability over my parameter space as my confidence coefficient. So, couple of things to note here. So, when I write a probability here what is random here is the interval random or theta is random. It is the random quantity is this not the theta here theta is fixed ok. You should be clear like earlier when we are studying the basic probability we said what is the probability that x belongs to let us say some interval a and b. There let us say if let us say x is uniform let us say 0 1 and if I take a equals to 0.1 and b equals to let us say 0.5. So, a and b were given there and x is my random quantity and I was looking into and how I computed this probability I just computed this probability like 1 between 0.1 to 0.5 right for the uniform distribution this is how I computed. But here theta is fixed what is changing is L of x and U of x depending on your sample ok. And whenever I write this we actually mean that probability that my theta is L of x and U of x is a meaning ok. Often this interval estimators we have along with their along with their measure of confidence one confidence measure we have is the confidence coefficient together with that they are called as confidence interval ok. You need to first give me confidence interval and you have to tell me the measure of the confidence how you are going to measure the confidence maybe one I have defined one measure here I am interested in the worst case like the smallest value of my coverage probability. Maybe somebody has other way of measuring it ok for example I may just say that ok I am other possibilities I may say that theta is itself is coming from some distribution ok I do not know what distribution it is and then I may be interested in expected value of this probability. So, here you are making multiple things this interval is random and you will compute this quantity for a given theta, but then you want to take expectation over all possible values of theta you get that could be another measure of your confidence, but you will not get into that you will be only interested in the confidence coefficient here which is the worst case coverage probability together when you have this interval estimator with this confidence coefficient measure that has given to you you are going to take it as a confidence intervals ok. Let us see that this confidence coefficients I have of course, does this quantity depend on theta yes or no does this confidence coefficient depends on theta how should it depend on theta you have taken among all the thetas you are taking already smallest value of that right you are already taking infimum of them then where is the theta coming into picture here right. So, these are probabilities like see that probability is going to be between 0 1 right and you are looking into the smallest value of that probability ok. So, and you are taking the smallest value across all the set like you are already looked into all possible values of theta that theta set the parameter set. So, it will not depend on any particular theta here ok. So, then let us look into the coverage probability itself how coverage probability should definitely depend on the theta right because this is for a given theta whereas, your confidence coefficient is over all possible values of theta. So, let us see how to how the theta affects our coverage probability and should it depend on how we define our interval estimators. So, here is one example of calculating this thing on some estimators. Suppose let us say you have a random sample which is drawn from any form distribution with parameter 0 and theta and theta is obviously unknown. So, what could be a potential estimator for theta? So, I have written here one natural estimator for theta is simply take the max value of that can this be a good estimator for theta, but that is still a point estimator right why here is a point estimator. So, now let us try to come up with some couple of interval estimator based on this ok. So, what I have done here basically let us say this is my x 1 and I let us say I have already ordered them x 1, x 2, x 3 and let us say x n actually I should have put them in the in terms of our order statistics notation it is the notation right x 1 is the first order statistics x 2 is the second order statistic and now what I am taking is I am taking this to be the value of theta in the first case and I am calling it as y, but instead of that what I will do is ok now tell me this y will be most of the time can it exceed theta or it will be lesser than or equals to theta it is going to be less than or equals to theta right because maybe my my range is this much 0 to theta and the samples I have observed here and maximum value y I am going to get is somewhere here. So, thinking along those lines maybe I know that whatever the y I get my actual value should be slightly larger than that ok. So, keeping that in mind maybe I can take my intervals to be something here A y where A is something greater than 1 ok to the right of this and there may be another region let which we call it as B y I am now defining this to be my possible one estimator. Does this interval estimator make sense? Right now I am not telling what is A and B, but let us say A is just simply greater than 1. So, that I am scaling whatever the y I get by that number and look something on the other side also this is one my possible interval estimator. So, then white scale them you can offset them also. So, for that I will consider another possible estimator instead of scaling like this one I am just taking that to be y plus c and maybe y plus d I am just offsetting them ok. And here naturally I want this c to be greater than or equals to 0 and d of course, has to be greater than c, but maybe one more constraint we can put that as d has to be less than theta, d cannot be itself larger than theta ok. I mean we will be interested in d where it is way less than the value of theta ok. Now, let us try to compute what is the coverage probability of these two estimators we have ok. So, this is estimator 1. So, probability that theta belongs to A y B y interval this I can rearrange it as sorry this should have been capital y I have rearranged it by dividing them and I will get this. So, y is max of x 1, x 2 up to x n right. If x 1, x 2 are all uniformly distributed do we know the distribution of y? Did you people compute it before? What is the PDF of y? n into ok I just writing it here this is going to be by theta n if y is between 0 and theta and 0 otherwise ok. And then t is y by theta what will be its distribution what will be the distribution of t just by scaling right you can again check that I am just is going to be n minus t if t is between 0 and 1 and 0 otherwise ok after by scaling this. Now, can I compute this probability t I already have its distribution this is going to be integration of t which is n t n minus 1 between 1 by B and 1 by A dt this is going to be what 1 upon. So, you know how to integrate this right this is going to be what simply 2 to the power t to the power n and when I put t equals to 1 by A I am going to get this quantity n minus 1 by B which is n. Did I make it correct? Let me cross check that b 1 by A lower limit is 1 by A. Is this can you do a sanity check B is this is always going to be positive right because B is greater than A. So, this quantity is going to be smaller than this quantity and so this is fine. Now, notice that this quantity here does it depend on theta? This quantity does not depend on theta. So, in this case the confidence coefficient in this case it does not depend on theta irrespective of what is theta this is always you are going to get. So, confidence coefficient here is sufficient for interval estimator 1 is simply this quantity 1 by A to the power n 1 by B to the power n ok. Now, let us look into estimator 2 interval estimator 2. So, what is that that is going to be P of theta, theta belonging to y plus c comma y plus d. So, this is nothing, but probability that y theta is less than or equals to y plus d less than or equals to y plus c this further I want to simplify such that should I get it in terms of I think it is again try it better to get it in terms of t so that ok. So, let me divide it by t. So, this is going to be dividing here could be a issue what we can do is. No, we can do that rest of the minute ok let us not worry about time being theta what was our we said uniformly between theta what is the parameter space of theta did we say anything about where is the theta we just say theta is a theta like range. So, theta is a positive quantity here right because we said uniform 0 theta right. So, theta has to be naturally greater than 0 then only it makes sense. So, let us find then let us it is no issue in dividing by theta throughout c by theta less than or equals to 1 less than or equals to y by theta plus d by theta this is fine no issue in dividing by theta. So, now, let us get into this into y by theta. So, now, if I want to write y by theta here. So, now, I know that y by theta is upper bounded by 1 minus c by theta and lower bounded by 1 minus d by theta and I already know what is the distribution of y by theta it is this. So, this is nothing, but now the lower limit is 1 upon d by theta 1 upon c by theta and the distribution I have is t into sorry n into 2 to the power n minus 1 dt. So, what this is going to give me? This is going to give me 1 upon c by theta minus 1 upon d by theta to the power n. Now, you see that this quantity interval estimator 2's coverage depends on theta ok. Now, let us do couple of sanity checks ok. Now, let us take estimator 1 if I let n go to infinity what should happen to my coverage probability of estimator 1 if I let n go to infinity. So, ideally like if I have large number of samples I should be able to do a better estimate right large number of samples that I should be able to give you I should be able to exactly give an interval which covers and I should be more get more and more precise. So, as n tends to infinity this term goes to 0 this term goes to 0 because both of them are less than 1 1 by a is less than 1 1 by b is also less than 1. So, coverage probability goes to 0 for as this is as n tends to infinity as and for 2 for 2 does this happen for the second estimator also. So, as n tends to infinity this is also going to happen because theta is anyway less than d it has to be less than c. So, both of them are less than 1. So, this will also go as n goes to infinity, but now this is does not matter what happens to theta, but on the way if you look into this as theta goes to infinity what is going to happen to the coverage probability of estimator 2 this is also going to go to 0 right because if you fix c and d and let theta go to infinity this get killed this get killed then 1 minus 1 is going to be 0. So, second one irrespective of where is theta if you have enough number of samples it will also give you a good coverage probability. So, let me stop.