# Statistics Gamma Distribution question

1. Oct 6, 2011

### stevenham

1. The problem statement, all variables and given/known data
Let X have a gamma distribution with parameters α and β.
Show that P(X ≥ 2αβ) ≤ (2/e)2

2. Relevant equations

f(x) = pfd of a Gamma

3. The attempt at a solution

I began by solving for P(X ≥ 2αβ) by doing ∫ f(x) dx from 2αβ to ∞
I set y=x/β for substitution.
and I got up to 1/$\Gammaα$ * ∫yα-1 e-y dy from 2αβ to ∞

I don't really know what to do from here.
I know that ∫yα-1 e-y dy from 0 to ∞ = $\Gammaα$
but I'm kind of lost because of the 2αβ

I am not even sure if this approach to solving this problem is correct.

For the second part of the question, we have a theorem that says :
P(X≥ α) ≤ eαtM(t)

My guess for the second part is to simply solve for eαtM(t)
Is that correct?

Any help will be greatly appreciated. Thank you.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution