Relation between Gamma and Poisson

[lily_w]

I'm having trouble doing a classic proof (integration par part and induction on r) for this :

Pr(X>t)=Pr(Y ≤r−1), where X follows a gamma Γ(α = r, β = 1/λ) and Y a Poisson P (λt).
Start with r = 1 (exponential distribution).

I don't really understand what induction on r really means.

I tried just showing the basic density equation for both (Gamma and Poisson) but i don't know how to make them simplify into the same thing.

 

[vela]

First, you want to show that the statement Pr(X>t) = Pr(Y≤r-1) is true when r=1. This is the base case. Next, you assume the statement holds when r=k and show that the statement is true for r=k+1. That's what you need to do for a proof by induction.

When r=1, what are the probability density functions for X and Y? Write down expressions for Pr(X>t) and Pr(Y≤0) and show they are equal.