 here the theorem 2, the multivariate normal, now next we have theorem 2, what is it saying? that last we have done, in that we have checked that there will be transformation, linear transformation, so we have transform variable also, we are following linear transformation. transform variable also will follow the same which is the original variable. here suppose the x follows the multivariate normal distribution with mean vector mu, various covariance matrix sigma. also let's see, c is the p cross p matrix, we have positive definite matrix such that c into c inverse which is equal to sigma, variance to various matrix. here suppose z equals to c prime c inverse x minus mu, this is also the transformation we are seeing here, now what is the transformation? x follows the multivariate normal distribution, so transformed variable also follows the multivariate normal distribution. so taking the expectation, c inverse expected value is applied to us on this factor, i.e. this random variable is here, we applied expectation on it, c inverse as it is, expected value of x minus mu. so we know that the expected value of x which is equal to mu, mu minus mu is cancelled out, so the expected value of z which is equal to zero. further the variance of transformed variable we have checked here, the variance of the transformed variable which is equal to expected value of z z prime. so z which is equal to c prime x minus mu into z transpose x minus mu transpose into c inverse transpose expected value of this this is the equals to variance to variance matrix because you have covariance, variance to variance matrix, sigma, we have entered here, so c inverse sigma c inverse so sigma which is equals to c c prime this is the c c prime which is equals to identity identity matrix is p cross p cap c c prime identity matrix c inverse c inverse transpose identity so this total we have equal to identity matrix so variance covariance matrix whose answer we have p cross p key identity matrix so this is the z transformed z follows the standard normal multivariate distribution this is the standard normal multivariate distribution so f of z equals to 2 pi minus p by 2 expected value of this this is the pdf of the transformed distribution transformed variable now we have z which is equals to c inverse x minus mu so x value we have determined so what will happen to x so what will happen here then plus mu x value and the Jacobian we have found the Jacobian so Jacobian curly z over curly x further we have solved it so z over curly x so what will happen here mod of determinant of c inverse so mod of determinant of c inverse which is equals to c c prime modulus minus 1 by 2 and this is equals to modulus of i.e determinant of variance covariance matrix i.e Jacobian variance covariance determinant of variance covariance matrix so z prime z z prime we have laid supposition which is equals to this one z which is equals to this one now further you have this transpose so transpose first comes this values then transpose is taken c inverse as it is x minus mu as it is this factor as it is this is also inverse so we can take it in one inverse c transpose c c transpose inverse this value we have taken x minus mu so we know that c c prime c transpose which is equals to variance covariance matrix sigma previously i have written that this factor is equal i have entered the value of inverse sigma t c transpose so the required probability density function of multivariate normal distribution is this f of x which is equals to this you have which one we have this is same pdf which we have multivariate normal distribution pdf same we have same pdf i.e it transformed variable we have follows the multivariate distribution if the original variable follows the multivariate normal distribution