Recall that a standard normal random variable has a expected value / mean of 0 and a variance of 1. Then a random vector of standard normal is denoted as . Where is the random vector with a size of , is a vector of zeroes, and is a identity matrix.
Suppose there are independent standard normal random variables then their joint density is the product:
Then the joint density can be simplified to
Where is a vector that contains , and is the dot product of itself.
The expected value of the random vector composed of standard multivariate normal is a vector of zeroes:
For random variables, the covariance matrix is an identity matrix.
Given that is the random vector of standard multivariate normal; any multivariate normal random vector can take the form:
Where is the random vector comprised of random variables with general normal distribution; vector has a length of . is an invertible matrix. And is a vector of constants.
Because is invertible, then
The mean of a generate multivariate normal is since the mean for the standard multivariate normal is :
The covariance is where is the transpose of :
Notice that is in fact a transformation, therefore its corresponding Jacobian is simply
Now that we know the Jacobian, it follows that the density is given by
Plugging in , and we obtain
Note that the covariance matrix , which also implies .
Also note that the determinant of the covariance matrix is , working it out we see that .
Plugging these equations in, the above density simplify down to
1. Linear transformation of normal vectors results in normal vectors
Suppose we have a vector , and a matrix is full rank. then let . The mean of is ; the variance is .
2. Marginal distributions are normal
Suppose we have a random vector that has size 2:
Then we get
3. Conditional distributions are normal
Using the previous case, suppose we have as a realization for , then the conditional on is
Where and .