Finding the Joint PMF of Two Independent Poisson Random Variables

[chili237]

Homework Statement

X~Pois(λ)=> p_x(k)=e^-λλ^k/k!

Y~Pois(μ)=> p_y(k)=e^-μμ^k/k!

Find p_X,X+Y(k,n)=P(X=k, X+Y=n)

Homework Equations

...I know the pmf for X+Y ~ Pois(λ+μ)

The Attempt at a Solution

As I understand the joint pmf for two independent random variables would be the product of the two individual pmfs. However as X+Y is dependent on X I got really stuck trying to think about this one and how to set it up.

Any help would be great. Thanks :)

 

[Office_Shredder]

If X=k and X+Y=n then Y=?