 Dear student, here is the theorem. Theorem of subset of multivariate normal. So, subset of multivariate normal, this is the statement of the theorem. If x1, x2 up to so on xn have a joint distribution, meaning x1, x2 up to so on xn, we have a joint distribution, we have a joint distribution, a necessary and sufficient condition that one subset of a random variable and the subset of random variable be independent is, what will happen, independent case is that each covariance of variable from one set and the variable from the other set may be zero. Now, I am explaining this theorem's little bit of a statement. This is the x1 and this is the x2. Now, we are saying that the necessary and sufficient condition then the one subset, this is the subset. We have this total x, we have this subset. This is the subset of a random variable, this is the subset of the random variable and the subset of the random variable be independent. Now, the subset that we have is independent from the other subset. It means that here you have the covariance term that will be zero. Means, xij, the covariance term that we are taking, x1, xi, x2, xj, the covariance term that you have will be zero. We have zero subset covariance, so we are saying that we have this set and this set, their covariance of variable will also be zero. So, this is the statement of the theorem that we have to do further. If xi is from one set and xj is from the other set, from any probability density function, sigma ij is defined as sigma ij you have covariance, covariance which is equals to expected value of xi minus mu into other set xj minus mu. This is the covariance, discrete. I have the covariance value. When we come to continuous now, the expectation is open. Minus infinity to infinity up to so on. Minus infinity to infinity, xi minus mu into xj minus mu. This is the density function, probability density function. So, sigma ij which is equals to integral minus infinity to infinity. What did we do further in this? We separated the ith term and separated the jth term. This is for the ith term. Subset, right? As I told you in the previous video, xi means we have to take values of x1 and second we have values of x2. So, this is the ith term and this is the jth term. So, sigma ij which is equals to expected value of xi minus mu, mu i is not a vector, xi minus mu. This term which is equals to xi minus mu and the remaining term which is equals to xj minus mu j, which is equals to zero. So, sigma ij, you have basically covariance which is equals to zero. Now, find the correlation. If we have to see its correlation, if the covariance is zero, then you know that you have the correlation which is zero. Hence, you know that this is the definition of the correlation which is equals to covariance of the covariance divided by standard deviation of ith and the standard deviation of the jth. So, covariance, you have zero. First, this is the value of zero divided by sigma i, sigma j. So, this is the correlation of ij. If you have covariance zero, then correlation which is also equals to the zero. So, this is the theorem that you have covariance. If the covariance of the subset is zero, then you have the overall individual variable. If the covariance is zero, then the covariance of the subset is also equals to zero.