Calculating the Joint PMF of Two Independent Poisson Random Variables

[chili237]

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.

 

[chili237]

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.