Solved – Sum of Normally Distributed Random Variables

normal distribution

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

I am aware that the sum of two (or more) normally distributed Random Variables is not necessarily also normal.

Yes, it's usually only the case if they're jointly normal (multivariate normal)

For c ∈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?

Yes.

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)?

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.

Related Question