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Find joint pmf of X and X+Y

  • Thread starter chili237
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  • #1
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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 :)
 

Answers and Replies

  • #2
chiro
Science Advisor
4,790
131
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.
 
  • #3
3
0
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.
 

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