Statistics Gamma Distribution question

[stevenham]

Homework Statement

Let X have a gamma distribution with parameters α and β.
Show that P(X ≥ 2αβ) ≤ (2/e)²

Homework Equations

f(x) = pfd of a Gamma

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/$\Gammaα$ * ∫y^α-1 e^-y dy from 2αβ to ∞

I don't really know what to do from here.
I know that ∫y^α-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) ≤ e^αtM(t)

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

Any help will be greatly appreciated. Thank you.

 

[mighty2000]

Thank you for your post. I am a scientist and I would be happy to help you with this problem. Your approach to solving this problem is correct so far. To continue, you can use the property of the gamma function which states that Γ(x+1) = xΓ(x) for any positive real number x. This will allow you to simplify the expression in the integral and solve for P(X ≥ 2αβ).

For the second part of the question, your approach is also correct. You can use the theorem you mentioned to solve for eαtM(t). Remember to substitute the appropriate values for α and β in the expression.

I hope this helps. If you have any further questions, please don't hesitate to ask. Good luck with your problem!