I looked at normal random variables in an earlier post but, what does it mean for a sequence of real-valued random variables to be jointly normal? We could simply require each of them to be normal, but this says very little about their joint distribution and is not much help in handling expressions involving more than one of the at once. In case that the random variables are independent, the following result is a very useful property of the normal distribution. All random variables in this post will be real-valued, except where stated otherwise, and we assume that they are defined with respect to some underlying probability space .
Lemma 1 Linear combinations of independent normal random variables are again normal.
Proof: More precisely, if is a sequence of independent normal random variables and are real numbers, then is normal. Let us suppose that has mean and variance . Then, the characteristic function of Y can be computed using the independence property and the characteristic functions of the individual normals,
where we have set and . This is the characteristic function of a normal random variable with mean and variance . ⬜
The definition of joint normal random variables will include the case of independent normals, so that any linear combination is also normal. We use use this result as the defining property for the general multivariate normal case.
Definition 2 A collection of real-valued random variables is multivariate normal (or joint normal) if and only if all of its finite linear combinations are normal.