# Thread: chi square degrees of freedom

1. ## chi square degrees of freedom

If X and Y are independent random variables, X having a chi-square distribution with 3 degrees of freedom, and X+Y having a chi-square dist with 7 degrees of freedom, prove that Y has chi-square dist with 4 degrees of freedom

Thank you for any help that can be provided

2. I would use MGFs.

use $\displaystyle X\sim\chi^2_3$ and $\displaystyle W=X+Y\sim\chi^2_7$

Then $\displaystyle M_W(t)=M_X(t)M_Y(t)$

Plug in and by division you have the MGF of Y.

3. How do you know (aside from practice?) when to use mgf vs. anything else? We just looked at a getting the limiting distribution of

$\displaystyle \frac {X_r-r*\beta} {\sqrt r*\beta}$

where

$\displaystyle X\sim\gamma(r,\beta)$

and the solution was best obtained using mgf. Thanks!

PS: I was able to do the problem, but had the hint do use mgf, so just trying to get a better sense of how to tackle these problems without hints.

4. That's just the central limit theorem

$\displaystyle \Gamma(\alpha,\beta)$ is equal in distribution to

$\displaystyle \sum_{i=1}^{\alpha} \Gamma_i(1,\beta)$