Can someone please explain why the following two scenarios lead to different distributions?
1) $X\sim\chi^2(m)$ and $Y\sim\chi^2(n)$. Using moment-generating functions, show that $Y-X$ does not have a $\chi^2(n-m)$ distribution, for $n>m$ with $X$ and $Y$ independent.
Refer to the following for a derivation of this: Distribution of Difference of Chi-squared Variables
2) Show using MGF's that if $X\sim\chi^2(m)$ and $S = Y+X \sim\chi^2(n-m)$, then $S-X\sim\chi^2(n)$.
Here is my derivation:
$$
E[e^{(S-X)t}]=E[e^{St}]E[e^{-Xt}]\\
=\frac{M_S(t)}{M_X(t)}\\
= (1-2t)^{-(m+n)/2}(1-2t)^{m/2}\\
= (1-2t)^{-n/2}
$$
My question is: why can't we use this type of derivation (putting $M_X(t)$ in the denominator instead of using $M_X(-t)$) in (1)? Or alternatively, is there another way to show this using $M_X(-t)$ in the derivation?
Thanks!
Best Answer
Your mistake is that you want $E[1/Z]$ to be $1/E[Z]$ for a random variable $Z$ (in fact, when $Z > 0$ they are never equal unless $Z$ is almost surely constant).