1. help with probability proof

hi, i need help with this one,
by using Moment-generating function, show that :

thanks to all!

2. I'm not sure what the MGF has to do with this.
If you compare the two densities you can see that the Chi-Square is a particular gamma.
You can do the same thing with the MGFs, if you wish.

3. Hello,

hi, i need help with this one,
by using Moment-generating function, show that :

thanks to all!

$\chi^2_n$ has the MGF $G(t)=\frac{1}{\sqrt{(1-2t)^n}}=\frac{1}{(1-2t)^{n/2}}$

$\Gamma(n,\lambda)$ has the MGF $G(t)=\left(\frac{\lambda}{\lambda-t}\right)^n$
so $\Gamma(n/2,1/2)$ has the MGF $G(t)=\left(\frac{1/2}{1/2-t}\right)^{n/2}=\left(\frac{1}{1-2t}\right)^{n/2}$

and this is the same as above

4. first of all thanks for helping!

but...

the MGF for the gamma distribution (according to wikipedia) is:

and not as you mentioned...
and in this case the equaliy holds only for

5. There's sometimes a problem, as to know whether the second parameter of the Gamma distribution is $\theta$ or $\frac 1 \theta$