Third Moment of a Sum of Normal and Gamma

Question

I just ran into the next problem: The random variables $X$ and $Y$ are independent, where $X \sim Normal(1,1)$ and $Y \sim Gamma(\lambda,p)$ with $E(Y) = 1$ and $Var(Y) = 1/2$ How do we find $E(X+Y)^3$ ?? I've tried a convolution, which leads to a really ugly looking integral from which I then have to get the third moment. I've tried characteristic functions and ran into the same problem, I'm sure there has to be some other easy way to solve this. Any ideas?

Answer 1

Write $$\mathrm E\left[(X+Y)^3\right]=\mathrm E\left[X^3+3X^2Y+3XY^2+Y^3\right]$$ and use the fact that $X$ and $Y$ are independent, i.e. $$\mathrm E\left[X^pY^q\right]=\mathrm E\left[X^p\right]\mathrm E\left[Y^q\right].$$

Answer 2

$$E[(X+Y)^3] = E[X^3+3X^2Y+3XY^2+Y^3] = E[X^3]+3E[X^2]E[Y] +3E[X]E[Y^2]+E[Y^3]$$ when $X$ and $Y$ are independent.