So this is just by the way equal to the product

i = 1 to n of 2 pi to the minus 1/2,

e to the negative zi squared over 2.

Okay, so, we get the multivariate normal distribution just as the product of

a collection, the multivariate standard normal distribution just is the product

of a collection of IID standard normals.

So that's for an n by 1 vector z.

And then we might define a multivariate non-standard normal, say x.

So it would not necessarily mean zero in variance/covariance matrix is I.

We might define that as mu plus sigma to the 1/2 times z.

Where sigma is a variance/covariance matrix, sigma to the 1/2 matrix,

that decomposition is called the Cholesky decomposition.

So notice if we write x this way, the expected value of x is equal to mu,

because the expected value of z is 0, and

the variance of x is equal to sigma to the 1/2 variance of z,

variance of z, sigma to the 1/2 transpose.

Which variance of z is I, so we just use this fact and it equals sigma.

Okay, so using that we can define this non-standard normal,

multivariate normal distribution.

Which people would then just call then the normal distribution or

the multivariate normal distribution maybe.

And we would write that x is normal, mu, sigma.

Now, if the density is associated with it,

we could use the transformation to figure out the density associated with it.

And it is 2 pi, to the -n/2,

determinant of sigma to the -1/2,

e to negative x minus mu transpose sigma

inverse x minus mu divided by 2.