Given X,Y independent continous variables, $X,Y \sim \mathcal N(0,16)$, what is $X - Y$?

Question

Knowing that $X,Y \sim \mathcal N(0,16)$ then $X - Y \sim \mathcal N(0,8)$ or $X - Y \sim \mathcal N(0,32)$?

How should I know this?

Answer 1

There are some facts which are well known, which you can use:

1. Linearity of Expectation:

In your case it is $\mathbb E(X-Y)$=$\mathbb E(X)-\mathbb E(Y)$

2. If two (or more) random variables are independent, then the variance of the sum/difference of these random variables is equal to the $\underline{\textrm{sum}}$ of the variances of these random variables. In your case it is

$Var(X-Y)=Var(X)+Var(Y)$

Answer 2

Hint.

First, since $X$ and $Y$ are independent then nndeed $X-Y$ is normally distributed (why ?). Now, what are $\mathbb E[X-Y]$ and $Var(X-Y)$ ?