If $X$ and $Y$ are normally distributed random variables, what kind of distribution their sum follows?

Question

I was reading this question. It is about notation but I would like to ask something regarding the sum of two normally distributed random variables. If $X$ is a normally distributed random variable with mean $\mu_{X}$ and variance $\sigma_{X}^{2}$, and $Y$ is a normally distributed random variable with mean $\mu_{Y}$ and variance $\sigma_{Y}^{2}$, what kind of normal distribution does $X+Y$ follow when

• $X$ and $Y$ are independent,
• $X$ and $Y$ are uncorrelated,
• $X$ are $Y$ are correlated?

How does one give a proof for each case?

Answer 1

Regardless of whether $X$ and $Y$ are normal or not, it is true (whenever the various expectations exist) that \begin{align} \mu_{X+Y} &= \mu_X + \mu_Y\\ \sigma_{X+Y}^2 &= \sigma_{X}^2 + \sigma_{Y}^2 + 2\operatorname{cov}(X,Y) \end{align} where $\operatorname{cov}(X,Y)=0$ whenever $X$ and $Y$ are independent or uncorrelated. The only issue is whether $X+Y$ is normal or not and the answer to this is that $X+Y$ is normal when $X$ and $Y$ are jointly normal (including, as a special case, when $X$ and $Y$ are independent random variables). To forestall the inevitable follow-up question,

No, $X$ and $Y$ being uncorrelated normal random variables does not suffice to assert normality of $X+Y$. If $X$ and $Y$ are jointly normal, then they also are marginally normal. If they are jointly normal as well as uncorrelated, then they are marginally normal (as stated in the previous sentence) and they are independent as well. But, regardless of whether they are independent or dependent, correlated or uncorrelated, the sum of jointly normal random variables has a normal distribution with mean and variance as given above.

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In a comment following this answer, ssdecontrol raised another question: is joint normality just a sufficient condition to assert normality of $X+Y$, or is it necessary as well?

• Is it possible to find marginally normal $X$ and $Y$ that are not jointly normal but their sum $X+Y$ is normal? This question was asked in the comments below by Moderator Glen_b. This is indeed possible, and I have given an example in an answer to this question.

• Is it possible to find $X$ and $Y$ such that they are not jointly normal but their sum $X+Y$ is normal? Here, we do not insist on $X$ and $Y$ being marginally normal. The answer is Yes, and an example is given by kjetil b halvorsen. Another, perhaps simpler, answer is as follows. Consider $U$ and $V$ be independent standard normal random variables and $W$ a discrete random variable taking on each of the values $+1$ and $-1$ with probability $\frac 12$. Then, $X = U+W$ and $Y=V-W$ are not marginally normal (they have identical mixture Gaussian density $\frac{\phi(t+1)+\phi(t-1)}{2}$), and so are not jointly normal either. But their sum $X+Y = U+V$ is a $N(0,2)$ random variable.

Answer 2

Let $Z = X + Y$ be, when X and Y are independent you can prove it using the moment-generating function

$ M_{X,Y}(t) = E(\exp(t(X+Y)) \\ M_{X,Y}(t)= E(\exp\{tX\}\exp\{tY\}))\\ M_{X,Y}(t) = M_x(t)M_Y(t) \\ M_{X,Y}(t) = \exp\{ (\mu_X + \mu_Y)t + \frac{\sigma^2_X + \sigma^2_Y}{2}t^2 \} $

Then, $Z \sim N( \mu_X + \mu_Y, \sigma^2_X + \sigma^2_Y )$