Just need to check the answer for the following question:
Question
Suppose $X$ and $Y$ are two independent standard normal variables:
$X$ ~ $N(0,1)$$X \sim \mathcal N (0,1)$
$Y$ ~ $N(0,1)$$Y \sim \mathcal N (0,1)$
What is the distribution of $X + Y$ ?
My Working
$X+Y$ ~ N($\mu_1 + \mu_2$$X+Y \sim \mathcal N ( \mu_1 + \mu_2$, $\sqrt{\sigma_1^2 + \sigma_2^2})$$\sqrt{\sigma_1^2 + \sigma_2^2}) $
$X+Y$ ~ N($0 + 0$$= X+Y \sim \mathcal N (0 + 0$,$\sqrt{1^2+1^2}$$\sqrt{1^2+1^2}) $
$X+Y$ ~ N($0$, $\sqrt{2}$)$= X+Y \sim \mathcal N ( 0, \sqrt{2}) $
Does this look correct?
