Is the joint distribution of these two dependent Gaussian RVs, Gaussian?

Question

I have two dependent Gaussian variables $X_1,X_2$ with unit mean each, and standard deviations $\sigma_1=2a$ and $\sigma_2=a$ respectively, while $a>0$

Is the joint distribution of these two dependent gaussian variables also gaussian !?

noting that their dependence is in the shape $$X_2 = bX_1+X_3$$ where $X_3$ is another Gaussian RV with unit mean and constant variance. $X_1,X_3$ are independent

Please advise

Answer 1

The answer is yes. Note that two random variables $X, Y$ are joint Gaussian iff $aX + bY$ is Gaussian for all $a, b \in \mathbb{R}$.

Answer 2

Yes, in your case, the joint distribution of two Gaussian random variables will be Gaussian, but this is not generally true (as per the comments).

Using characteristic functions, one can show that $(X_1,X_2)$ is multivariate Gaussian, iff $a_1X_1+a_2X_2$ is univariate Gaussian for all choices of non-zero vectors $(a_1,a_2)\in\Bbb{R}^2\setminus\{0\}$.

Let $\Sigma$ be the covariance matrix of $(X_1,X_2)$, i.e. $\Sigma$ is the $2\times 2$-matrix, whose diagonal elements are the variances of $X_1$ and $X_2$, and whose off-diagonal elements are the covariance between $X_1$ and $X_2$.

Given that $X_1$ and $X_2$ are not fully correlated, the joint density of $(X_1,X_2)$ is $$f_{(X_1,X_2)}(\mathbf{x}) = \frac{1}{2\pi \sqrt{|\Sigma|}}\exp\left(-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^T\Sigma^{-1}(\mathbf{x}-\boldsymbol{\mu})\right),$$ where $\mathbf{x} = (x_1,x_2)$, $\boldsymbol{\mu} = (\mu_1,\mu_2)$ are the mean values of $(X_1,X_2)$, and where $|\Sigma|$ is the determinant of $\Sigma$.