# Gaussian random variables [duplicate]

Can some one help me point in the right direction or point to some resources that will help me prove that sum of two jointly distributed Gaussian r.v. with a given correlation coefficient is also a Gaussian r.v. Thanks

• 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... Apr 12, 2015 at 22:55
• Thank you. But is the proof this simple? Don't I have to show something more? Apr 12, 2015 at 23:04
• If I recall correctly, it was shown in Kendall's The Advanced Theory of Statistics. Vol. 1: Distribution Theory. Apr 13, 2015 at 0:02
• What is your definition of jointly Gaussian random variables? If you use the one stated in my previous comment, then you don't need to "show" very much more than what I wrote. If you start from the joint density, then there can be a lot of algebra involved. A proof starting from characteristic functions, instead of joint densities, is given in this answer written by @probabilityislogic Apr 13, 2015 at 13:14