Confusion about Jointly Gaussian

Question

1. From here, it says that, linear combination of two Gaussian distribution, are always Gaussians.

2. However, Let be standard normal and =±1 with probability 1/2 each, independently of . Let =. Then is also standard normal, but =+ is exactly equal to zero with probability 1/2 and is equal to 2 with probability 1/2.

But (2) contradicts with (1). Am I missing anythings?

Answer 1

The simplest case is when you take linear combinations of independent Gaussian random variables, these will be Gaussian. However, $X$ and $Y$ are not independent. A more general case in which linear combinations of correlated Gaussian random variables are Gaussian is if $X$ and $Y$ are bivariate Gaussian (aka jointly Gaussian). You can read more about this difference between (1) $X$ and $Y$ are both Gaussian and (2) $X$ and $Y$ are bivariate (jointly) Gaussian here: https://en.wikipedia.org/wiki/Multivariate_normal_distribution

Essentially, there are many ways for variables to correlated. Being jointly Gaussian random variables is a special way for Gaussian variables to be correlated that implies this property we've discussed. In fact, jointly Gaussian variables can actually be defined by this property that all linear combinations are normally distributed.

Answer 2

Linear combinations of jointly Gaussians (also known as multivariate Gaussians) are always Gaussian; however, X and Y are not jointly Gaussian. (One of the easiest ways to define a joint Gaussian $X$ is to say that $X = AZ + b$ where $A$ is some matrix, $b$ is some vector and $Z$ is a vector of i.i.d. Gaussians.)