 Dear student, this is the conditional distribution. So what is the conditional distribution? A conditional distribution is the distribution of a random variable or a set of random variable when the remaining set of random variable are known or fixed. This is the basic definition of the conditional distribution. Let the random vector x, x is the multinomial distribution, multivariate distribution with mean vector mu and various covariance matrix sigma. Here the partitioned as of x, this is the random vector x is the partition x1 and x2. So how many dimension of the x1? The dimension of the x1 which is equals to the q into 1. And how many dimension of the x2 which is equal to p minus q into 1? Its dimension we have determined in the previous lecture. So x1 and x2. Here mu which is equals to mu1 and mu2, this is not the vector and various covariance matrix is the sigma1, sigma2, this is the partition matrix. Then the conditional distribution of x1 and x2 is the multivariate normal with mean vector and various covariance matrix is this. Here is the mean vector. This is the conditional mean x1 given x2, given x2, given x2. The variable we have in the expectation, we see the notation of capital X. We can write capital X here. And the mean of the conditional when x1 given x2 which is equals to this one. And the variance of the conditional x1 given x2 which is equals to this. Here this is the variance covariance matrix, conditional variance covariance matrix. And the notation of the conditional variance covariance matrix which is equal to the sigma11.2. That means, instead of writing the variance, if we write this as sigma11.2, it means this is the conditional variance. Now conditional variance in general, we have this. Further we check. The conditional distribution of x1 given x2. This is the conditional distribution of x1 given x2. Here is the mean of the conditional distribution which is equals to this. And variance covariance which is equals to sigma11.2. Which implies the mean vector and the variance covariance matrix of the conditional distribution is this. This is the mean and this is the variance covariance matrix. This is the mean. This is the expected value of x1 given x2. And this is the variance covariance. This is the variance. In general, we are writing x1 given x2 and variance is equal to this. Now the density function of the conditional random variable is this. So, this is the density function of the conditional distribution when x1 given x2 which is equals to this value. Here, you know that the density function of this multivariate normal x minus mu, this is the mean, variance covariance matrix, variance covariance conditional, variance covariance matrix and x1 minus mean. So, in general, we have defined the density function here of conditional distributions. So, really, we have not proved this. We are just telling here that the density function of the conditional distribution is this. Similarly, we can show the mean vector and the variance covariance matrix of the conditional distribution is x2 given x1. Now, on which condition? x2 given x1. We have the conditional. It's a P minus Q variate. Now, look at this. This is the expected value of x2 given x1, the mean of the conditional distribution x2 given x1 which is equals to this one and the variance x2 given x1 which is equals to this one and this whole which is equals to sigma 22 dot 1. This is the notation of the variance x2 given x1 and the conditional density. This is the mean. This is the variance and the conditional density when x2 given x1 which is equals to this one. You know that this is the mean x minus mu, variance covariance matrix and the x minus mu. This is the transpose. This is the general case of the conditional distribution when x2 given x1. This is the special case if we set Q equals to 1. Just one dimension, when Q equals to 1, then x1 equals to x. Look at this. What did you have? Vector x1 because Q is 1. We didn't have dimension. We have Q equals to 1. It's a special case. We have the condition of the conditional distribution. x1 equals to x with variance sigma 11 which is equals to sigma 11. The variance of first component. This is the variance of the first component. Then about its expectation, and the expected value of x1 given x2, this is the conditional x1 given x2 which is equals to mu1. Now there is no vector here because we are finding for single. We have found for the first component. So, check here. This is the mu1. We have the remaining vector. Which is the conditional mean of one variable on the set of variable. What is the variance? x1 given x2. So, what we have first? Variance of the first component. This is the small sigma 11 and remaining you have vector form. So, we have a special case of the conditional distribution. This is the special case of the conditional distribution. This is the conditional distribution.