5
$\begingroup$

Say I have $X \sim N(\mu_1, \sigma_1^2)$ and $Y \sim N(\mu_2, \sigma_2^2)$, also $X$ and $Y$ are independent, then is the joint distribution of $X$ and $Y$ multivariate normal? I.e., $$\begin{bmatrix} X \\ Y\end{bmatrix} \sim N\left(\begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}, \begin{bmatrix} \sigma_1^2 & 0 \\ 0 & \sigma_2^2 \end{bmatrix} \right) $$

If so, why?

$\endgroup$
5
$\begingroup$

One characterization of multivariate normality that is often taken to be the definition is that the tuple $(X_1,\ldots,X_n)$ has a multivariate normal distribution if for every tuple $(c_1,\ldots,c_n)$ of constants (i.e. non-random scalars), the linear combination $c_1 X_1+\cdots+c_nX_n$ has a univariate normal distribution. If $X\sim N(\mu_1,\sigma_1^2)$ and $a$ is constant, then $aX\sim N(a\mu_1,a^2\sigma_1^2)$. If $aX\sim N(a\mu_1,\sigma_1^2)$ and $Y\sim N(a\mu_2,a^2\sigma_2^2)$ then the distribution of $X+Y$ is that of the sum of two independent normally distributed random variables. Its expected value can be found without knowing that they are independent; its variance can be found if you know only that they are uncorrelated. But the fact that the sum is normally distibuted relies on the assumption that they are independent.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.