MGF Techniques for Chi-Square Distribution on 2n Degrees of Freedom

[dim&dimmer]

Homework Statement

rvX has [tex] f(x) = \alpha \exp^{-\alpha x} , and \ W = 2n \alpha \overline {X}[/tex] defines a random sample from the distribution.
Use moment generating function techniques to show that the distribution of W is chi-square on 2n degrees of freedom.

Homework Equations

The Attempt at a Solution

Well...
Ive let [tex] \alpha = \frac {1}{\beta}[/tex], then [tex]f(x)[/tex] ~ [tex] exp(\beta)[/tex]
[tex]M_x(t) = (1 - \beta t)^{-1}[/tex]
mgf of W with w~chisquare(2n)
[tex] M_w(t) = (1 - 2t)^{-2v} [/tex]

I don't really know what to do after this. Any help appreciated

 

[statdad]

I'll use [tex] T [/tex] for the statistic of interest.

[tex]
T = 2n\alpha \overline X = 2 \alpha \sum_{i=1}^n X_i
[/tex]

You know the m.g.f. of each [tex] X_i [/tex] (they are iid). When you begin to calculate

[tex]
\int_{-\infty}^\infty e^{st} \, dt = E[e^{st}]
[/tex]

remember that [tex] t [/tex] is a sum, and use properties of exponents and expected values. You should wind up with a product that will lead you to the answer. (This all relies on the fact that the moment-generating function uniquely identifies the [tex] \chi^2 [/tex] distribution.)