# Find joint pmf of X and X+Y

X~Pois(λ)=> px(k)=e-λλk/k!

Y~Pois(μ)=> py(k)=e-μμk/k!

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

....I know the pmf for X+Y ~ Pois(λ+μ)

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 :)

chiro
X~Pois(λ)=> px(k)=e-λλk/k!

Y~Pois(μ)=> py(k)=e-μμk/k!

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

....I know the pmf for X+Y ~ Pois(λ+μ)

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 :)

The X+Y problem in general can be solved through the convolution theorem. The requirement is that all random variables in the summation (in this case they are X and Y but they could X,Y,Z,W as in X+Y+Z+W) be independent.

Do you know about the convolution theorem? If not have you made any attempts at the problem? If so could you please show them so we can help you.

I'm completely new to probability, so I'm learning as I go. The convolution theorem isn't something I know or have in any of my materials, but I'll do some research and hopefully that'll point me in the right direction.