# Gaussian random variables [duplicate]

• Many people take the definition of jointly Gaussian random variables as a collection of random variables such that $\sum_i a_iX_i$ is a Gaussian random variable for all choices of real numbers $a_i$. Thus, what you are asking for is a tautology for these people. Since \begin{align}E[X+Y]&= E[X]+E[Y]\\\operatorname{var}(X+Y)&=\operatorname{var}(X)+\operatorname{var}(Y)+2{\rho}\sqrt{\operatorname{var}(X)\operatorname{var}(Y)}\end{align} apply even for nonGaussian random variables, there is little left to do... – Dilip Sarwate Apr 12 '15 at 22:55