Distribution of random variables when combined I need help with this problem:
If $X$ and $Y$ are two independent random variables and are both standard normal, what is the distribution of $\frac{1}{2}(X^2+Y^2)$?
I think I start with $M(t)=e^{\frac{1}{2}t^2}$ but I'm not sure where to go from there.
Any help is appreciated. Thank you!

The mgf of $X^2$ can most easily be computed using the density:
$$ M_{X^2}(t) = E[\exp(tX^2)] = \dfrac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty
e^{-x^2/2 + t x^2}\; dx$$ 
The mgf of $X^2 + Y^2$ is then the square of this, and the 
mgf of $(X^2 + Y^2)/2$ is 
$$ M_{(X^2 + Y^2)/2}(t) = E[\exp(t(X^2+Y^2)/2)] = M_{X^2 + Y^2}(t/2)$$
Hopefully you may recognize the result.
