I am aware that the sum of two (or more) normally distributed Random Variables is not necessarily also normal.
I do have questions regarding the special case where I have the following addition:
For c $\in \mathbb R$, is c + N(0,a), where I add a normal Random Variable with mean zero and variance a to c, is then the sum normally distributed? Also, does this addition correspond to drawing a random variable X too from a Normal distribution with mean and then add it to N(0,a)?
Hope the question is clear, thanks
Best Answer
Yes, it's usually only the case if they're jointly normal (multivariate normal)
Yes.
You mean, if $Y\sim N(0,a)$ and $X\sim N(\mu_X,\sigma^2_x)$? The previous result means that $X+Y|X=c$ is normal. That's useless when you don't condition on the value of $X$, though, and then the form of the dependence between $X$ and $Y$ that you started with again comes in.