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


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