If $U$ and $V$ are independent identically distributed standard normal, what is the distribution of their difference?
I will present my answer here. I am hoping to know if I am right or wrong.
Using the method of moment generating functions, we have
\begin{align*} M_{U-V}(t)&=E\left[e^{t(U-V)}\right]\\ &=E\left[e^{tU}\right]E\left[e^{tV}\right]\\ &=M_U(t)M_V(t)\\ &=\left(M_U(t)\right)^2\\ &=\left(e^{\mu t+\frac{1}{2}t^2\sigma ^2}\right)^2\\ &=e^{2\mu t+t^2\sigma ^2}\\ \end{align*} The last expression is the moment generating function for a random variable distributed normal with mean $2\mu$ and variance $2\sigma ^2$. Thus $U-V\sim N(2\mu,2\sigma ^2)$.
For the third line from the bottom, it follows from the fact that the moment generating functions are identical for $U$ and $V$.
Thanks for your input.
EDIT: OH I already see that I made a mistake, since the random variables are distributed STANDARD normal. I will change my answer to say $U-V\sim N(0,2)$.