 Dear student, as we know the idea of multivariate analysis, the question arises how to find the descriptive statistics or summary statistics. Now the mean vector and covariance matrix in multivariate analysis, how to find the mean vector and the covariance matrix in multivariate analysis. Now the summary statistics says that the descriptive statistics are measured that describe the data in multivariate analysis. The most commonly used by are the mean vector, covariance matrix and correlation. Mean vector just mean find here, we have to find the mean vector in the multivariate. Covariance, we have to find the matrix. You may find the covariance in the univariate, but we have to find the covariance matrix and then find the correlation. An obvious extension of univariate notation, meaning obvious of extension of univariate means the univariate we have already found is the extension in multivariate analysis to find the mean variance. The i-th variable mean, the sample mean of the i-th variable, what will happen to us? Sample mean of the i-th variable, I mean we know what mean is in univariate. X bar, what do you have? Some x i over n. But here is the sample mean of the i-th variable that we have to find. So, what is the i-th variable mean? X bar i, 1 over n, some j varies 1 to n, x ij. Now j varies 1 to n, as you know we have the matrix. We have the i-th variable on the row side and we have the columns on the j side. So, i-j, multivariate analysis is the i-th and j-th we are taking here. So, here you will have x ij. So, the i-th variable for the sample mean is x bar i, some i varies 1 to n, some j varies 1 to n, x ij. This is the mean of the i-th variable. Same as the sample variance of the i-th variable, we have to find the sample variance of the i-th variable. Then, what is the notation we have? Basically, we are looking at the notation. We are defining it in the descriptive. Sigma ii. Now, what is sigma ii? You can also write sigma i square in this and we can also call it sigma ii. Now, sigma i square, you know that sigma square which is equals to variance and the sigma i, it means the sample variance of the i-th variable. Which is equals to 1 over n, some j varies 1 to n, x ij minus x bar i whole square. What was the simple variance you had? What was the simple variance you had? Sigma square which is equals to sum x i minus x bar sample variance whole square divided by, here you have small n. So, for sample variance, if we are taking it here, because sigma square's notation is sigma square which is equals to the population variance, so capital N was in the univariate case. So, here we are taking it, you have it in multivariate. So, here we will give the x bar from mu because this is the population variance. Here we are talking about the population variance. Okay, next is finally the sample covariance of the i-th and j-th variable. Covariance, you know the covariance between two variables we will check. Now, the covariance, notation is the sigma ij which is equals to 1 over n, sum i varies 1 to n x ij minus x bar i and the x ij minus x bar j. These are the mutations. Now, summary statistics individual. Now, individual, what do you mean by individual? What are the single variables we take? Means, in univariate, we have summary statistics. The vector of means x bar, x bar 1, x bar 2 up to so on x bar p is called the sample mean vector or simply the mean vector. Okay, simple, we have to find the mean vector of means. We have to find the vector of means. It represents the center of gravity of sample points, the P into P matrix. A P into P matrix, I have explained it last time. What is P into P matrix? We are giving the notation of sigma ii x vector. That is, here you have sigma ii basically, what will happen? We are finding it for single variable. With elements given, sigma ii and sigma ij is called the sample covariance matrix, that is, you have sample covariance matrix notation. Now, the covariance between x i and x j are sigma ik. Now, the sigma ik, which is equals to the integral, now you have i, if x i and x k are the continuous random variable, x we have a random variable. Now, x random variable, if we have continuous, then what we have to use is integral. So, then we use integral. And if we have discrete random variable, then we use summation. So, then we talk about this in sum. Now, next, the sum of the statistics is a P into n data matrix. Each element of x is a random variable and its own marginal probability distribution. You also know this is a univariate case. So, in univariate case, basically, what we have? In univariate case, mu i for the ith variable, mu i expected value of x i and the sigma i square expected value of x i minus mu i whole square, where i varies 1 to P. Similarly, x i we have is a continuous random variable. So, we use integral here. If x i is a discrete random variable, then we use summation sum. Similarly, sigma i square varies. Variance, x i is a continuous random variable. So, integral is used and x i is a discrete random variable. So, we use sum's notation here. These are all notations. We will use all these notations further in the analysis. So, the purpose of telling you here is that in some statistics, you basically know how to find the mean, what is the notation you have, how to find the variance, how to find the covariance, what are the notations we have. Now, we are using matrix form. Now, how will matrix be basically? The mean vector and the covariance matrix in multivariate. Using these notations when variable as column and into P data matrix. Now, if you go to data matrix, then what we did there? Variable as column. So, variable as column, then what is the notation we have? And if variable in rows, then what is the notation we have? In notations, there is a little difference. Now, using these notations when variable as column and into P data matrix. Now, mu vector, one over n, x vector transpose into one. What we have basically? Where one is an one in n vector of ones. I mean one, we have a vector. Basically, we have a vector of ones. And where does one go up to one? n times the one's vector we have. This is the notation of mean vector. Now, sigma, variance covariance matrix. One over n, x prime, x prime means transpose. I had told you last time that prime or we are using the sign of prime or we can use transpose here. If we apply t, then transpose will be used. Basically, prime means transpose will be used. Identity minus one over n, one into one transpose into x. So, we have the notation of variance covariance matrix. How we have to point variance covariance matrix? Where identity is n into n identity matrix. Now, what is the identity of n into n matrix? This is one in n vectors. And here, you have n into n identity matrix. Now, further, how is this solved? You can simply multiply this x. You can see that it has been multiplied. Then, you have this by-rex of our bracket. We have multiplied it. And further, we have results. As you know that identity matrix, if we multiply it with any matrix, then there is no change on it, it remains the same. Identity matrix has been multiplied. So, what is left from here? x prime x. Because identity matrix has been multiplied. Identity matrix has been multiplied by anyone. So, we do not have any effect in the results. Finally, we have the results here. Who will get this factor and who will get this factor? Here, we have written mu into mu prime. Now, x transpose 1. So, whose cake was x transpose 1? Mu cake will be 1 transpose x. Whose cake will it be? Which is equals to mu prime. So, this is the basic result of variance-covariance matrix. And when will we use this? When variable as column. We have to use it like this. Next, using the notation when variable as row. Now, where is the variable we have? Rows. If you remember in the data matrix, I had given two examples. So, we are solving it according to the second example. When we have the variable in rows. So, what will be the notation then? Mu mean vector which is equals to 1 over n, x into 1. Now, what was in previous? x prime. And here, we are taking x into 1. Where 1 is an n, vector of 1. Same way, you have variance-covariance matrix here. Where x prime was here. And here, x was here. Here, your change is done. Same way, you have multiplied x. Then, we have x prime multiplied. And after the solution, you check it, the solution is done. This result will come. We have to use this result when we have variable as rows. And in the previous result, we will find variance-covariance when we have variable as columns. Now, next, the summary statistics mean vector, covariance matrix and correlation matrix. Similarly, population mean vector is defined as. A population mean vector is defined as. Find, we have to do the same, summary statistics. But we have to find the mean vector, covariance matrix, correlation matrix. As we have already found. Similarly, population mean vector is defined as population mean vector. So, you know population may have. Simple population, what is mu? Mu, what is it? Some x over n, what is n? Capital n, population size. So, mean vector, population mean vector is defined as 1 over capital n. Some i varies 1 to n, x, i. Population covariance matrix, we have. Simple, you have. Variance-covariance matrix, 1 over n, some i varies 1 to n. X i minus mu, X i minus mu prime. Univariate case, we have. Some X i minus mu whole square. How we have written whole square? X i minus mu into X i minus mu transpose. Now, next, you solve it further. So, you solve it further, as we have opened square. Similarly, you have multiplied it with transpose. Multiply it. Some factors will cancel out. How will they cancel out? Plus minus sign will come. We will enter its value instead of mu. Then, those values will cancel out. After canceling out, you will have the final result. This will come. There are just two steps in this. After solving two steps, we will reach the final result. And the population correlation matrix. Population correlation. You know, how we define population correlation? From rho. Here, you have matrix. What will we have with rho matrix? D inverse. Now, what is D inverse? We have D inverse basically. They know diagonal of sigma ii square root. That is, we have D inverse. That is, what is D? Which is equals to sigma 11 square root diagonal. Sigma 22 square root of diagonal 0. So, we will have D. As he said, D which is equals to diagonal of sigma ii square root. So, we have D inverse. From here, we have D. Sigma, we have various matrix. And we have multiplied D inverse. So, we will determine the correlation matrix. Sample sum of square and product is defined as. Now, sample sum of square. As you know, the sum of square we had. Sum xi minus x bar whole square. Sample sum of square. Now, what is the method of writing this in the multivariate? We have taken the notation of capital A. Sum i varies 1 to n. xi minus x bar into xi minus x bar transpose. In previous transpose, there was this sign. Here, it had t as the notation. So, both of them. As I said, there is no difference in the notation. There is a difference in the notation. But the thing is, you can do the sign transpose with t or with prime. Now, further, you will multiply it like this. You have multiplied it like this. You have multiplied it like this. And there will be two factors in the center. They will cancel out. Now, the sample covariance matrix is defined as. Now, we have a sum of square. After sum of square, we have to find sample covariance matrix. Now, sample covariance matrix you have sum of square divided by small 1 over n. And the sum of square multiplied by 1 over n. What do you have? Sample covariance matrix. Further, we have it like this. We can say that we have to do algebra. We have to find the algebra in this calculation. After solving the algebra, we have final sample covariance matrix. This result will come. You can also use this result. Finally, we know that this factor is one factor which is equals to the a and a into 1 over n. This is the a sample covariance matrix. Finally, the sample correlation matrix is defined as population correlation matrix. We denote it as rho. And we denote the sample correlation matrix as r. Now, r which is equals to r matrix because the sample correlation matrix is r which is equals to d inverse s. As we have just found, sample covariance matrix and d inverse. Any d inverse into x into d inverse. Where d is equal to the diagonal of sigma ii. As we have just done. Now, we have to do a brief summary. In brief summary, we have to find everything. I have written it as a brief. You have to use all the values and notations in the idea of what you want to use. I have given a brief summary. We have the mean vector mu i. Expected value of x i. We use the integral value when x i is random variable continuous. And we use sum when we have random variable discrete. We have simple summary statistics. You have mu p into 1. What is mu p? P rows and one column. So, you have this. P rows and one column. So, the value of mu is mu 1, mu 2 up to mu p. Variance covariance matrix. Variance covariance matrix. We use this result. Sometimes, we will use the previous result. And sometimes, we will use this. But mostly, it depends on the nature of the analysis and the nature of the question. Which result we have to use.