Use MGF's to show that $S-X \sim \chi^2$ with $n$ degrees of freedom

Question

Suppose that $X \sim \chi^2$ with $m$ degrees of freedom, $S=X+Y\sim \chi^2(m+n)$ and $X$ e $Y$ are independent use MGF's to show that $S-X \sim \chi^2$ with $n$ degrees of freedom this is my work

Answer 1

You're on the right track. Proceed this way:

1. Start with the MGF of $S$.
2. Express the MGF of $S$ as the product of the MGFs of two RVs $X$ and $Y$
3. Given the distribution of $X$ pin down the MGF of $Y$.
4. Conclude that $Y$ has the desired distribution.