MGF and Correlation Let $X,Y$ be iid RVs with common standard normal density. Let $U=X+Y$ and $V=X^2+Y^2$. Find the MGF of the random variable $(U,V)$. Also, find the correlation coefficient between $U$ and $V$. Are $U$ and $V$ independent?
Based on sums and squares of Normal RVs, I would say that U is N(0,2) and that V is ChiSquare(2). Then the marginal densities are f(u)=$\frac 1{\sqrt {4\pi}}e^{-u^2/4}$ for $-\infty<u<\infty$ and f(v)=$\frac 12e^{-v^2/2}$ for $0<v<\infty$
Then I believe the Jacobian of this transformation is J=$\frac 1{2{\sqrt {2v-u^2}}}$ (not sure on this) which would then make the joint pdf f(u,v)=$\frac 1{\sqrt {4\pi}}e^{-v/2}(2v-u^2)^{-1/2}$
I'm not sure where to go from here or if this is even right.
 A: You don't need to calculate the joint density, you just need to calculate the moment generating function. Since $X$ and $Y$ are independent, the MGF factorizes in the following way:
$$E[e^{t_1U + t_2V}] = E[e^{(t_1 X + t_2 X^2) + (t_1 Y + t_2 Y^2)}] = E[e^{t_1 X + t_2 X^2}] E[e^{t_1 Y + t_2 Y^2}]$$
Now you just need to calculate the integral
$$\begin{align*}
&\frac{1}{\sqrt{2\pi}} \int \limits_{-\infty}^\infty \exp(t_1x + t_2x^2 - x^2/2) \, dx \\
={}& \frac{1}{\sqrt{2\pi}} \int \limits_{-\infty}^\infty \exp\left((t_2 - 1/2)\left(x + \frac{t_1}{2 \sqrt{t_2 - 1/2}}\right)^2 - \frac{t_1^2}{4t_2 - 2} \right) \, dx \\
={}& \frac{1}{\sqrt{1 - 2t_2}} \exp\left(\frac{t_1^2}{2 - 4t_2}\right)
\end{align*}$$
(The integral diverges for $t_2 \ge \frac{1}{2}$).
To get the mixed moments of $U$ and $V$, you need to consider the connection between the derivative of the MGF and the mixed moments. As for the independence, you just need to check if the moment generating function factorizes in the way that it does with independent random variables. 
