 Dear student, this is the theorem number one of the subset of normal distribution, subset of normal distribution. Up subset here, let the random vector x is normally distributed, multivariate normal distribution with mean vector mu and variance covariance metric sigma. Then all subset of x are normally distributed, that is, if the partition of x is mean vector mu and the variance covariance metric sigma, partition of x, here the x, the dimension of the x which is equals to pain to one and the subset of x is the x one and the x two. The dimension of the x one which is equals to q into one and the dimension of x two which is equals to p minus q into one. Now here, x is a vector with p into one dimension. Now the x one, x two up to so on, x q, the first subset x q. How many dimension? q dimension and the other value of the x q is the x q plus one up to so on, x p. This is the p into one dimension. So total am I repose kithiogi dimension p into one p rows and one column. This is the one set. This is the second set. So first set kithiogi dimension kithiogi dimension q, q rows into one column, second set. Second set kithiogi dimension kithiogi dimension p minus remaining up to p minus q how many column which is equals to one. This is the subset. The first subset kithiogi dimension q into one, second set kithiogi dimension p minus q into one. Is that you can see here develop where mu is the dimension of p into q and the mu one q into one and mu two p minus q into one and the various covariance matrix sigma. This is the variances and this is the covariances. Now the x one, this is the x one is distributed at multivariate normal with mean vector mu and variance covariance matrix sigma one one. This is the statement of the theorem. Now here the proof, let, now let am ne kya kye hai c, t am I repose basically kya hai known of the c is the known of the matrix, known hai iski values aur dimension ki itni hai q into p aur ham ne isko let kya hai this is the identity matrix and this is the zero matrix. Abhi identity case hai, zero case hai kyu case, t ki kithi dimensions hai, q into p, to x one, x two up to so on x q and other q am I repose kya hoga, x q plus one up to so on. Now next, x one q, x two q and the other x q q, ye matrix, this is the matrix aur ye matrix ko am ne kya hai, this is the identity matrix with how many rows and column, q rows and q column. This is the identity matrix, yo ham let kya hai, aur other than this, wu saare huma repose zero cakepul hai. So, this is the c, q into p dimension identity matrix and remaining q with zero, so that the z which is equals to c into x, so ham ne kya kye zi which is equals to c into x and we know that the value of c into x, x previous apko define hai, x one, x two. Now identity multiplied by x, so x one vector, zero multiplied by x two, zero. So, the z is followed the multivariate normal with mean vector mu and variance covariance matrix c sigma c prime, so c mu which is equals to, c which is equals to identity and zero and mu one, mu two, identity multiplied by mu one which is equals to mu one, zero multiplied by this. So, c mu which is equals to mu one, so c sigma c prime, c which is equals to this, sigma given hai hume and c prime, rose ko column mein convert kar di hai, wu apko paas aage. Now multiply the, these matrices, identity multiplied by c sigma one one and multiplied by identity which is equals to sigma one one. So, zero multiplied by sigma one two, into identity kya hoga zero, then identity multiplied by sigma two one, multiplied by zero, zero, zero multiplied by sigma two two, into identity zero. So, remaining hume hai repas iska, finally result kya aage c sigma c prime which is equals to sigma one one, yeh portion. So, z is x one, z kya hume hai let kya hai, which is equals to x one is normally distributed, normally distributed with mean vector mu one and variance covariance matrix, sigma one one. As required, this theorem state that all subset of multivariate normal are normally distributed. All subset means, x one agar folokar hai, normal distribution ko, x two amai repas folokar hai, multivariate normal distribution ko toh har factor uska folokar hai, normal distribution ko.