Finding the PDF of the cubed difference between two independent standard normal random variables

Question

Let $X_1$ and $X_2$ be independent $\operatorname{N}(0,1)$ random variables. Find the PDF of $(X_1-X_2)^3 / 2$ .

I see that if it were the squared difference then I could use chi-squared with $1$ degree of freedom—but this one is a cubed difference. Can I follow the same approach? Any suggestions appreciated!

Answer 1

If $Y=X_1-X_2$ then $Y$ is normal with mean $0$ and variance 2, so you can write down the density of $Y$. Can you now compute that density of $\frac {Y^{3}} 2$?.