Geometric Distribution, Poisson

AI Thread Summary
The discussion revolves around calculating Pr(M>1) given that N has a geometric distribution and M has a Poisson distribution, with the conditions E(N) = E(M) and Var(N) = 2Var(M). The expected value and variance formulas for both distributions are provided, leading to the relationship λ = (1-p)/p and 2λ = (1-p)/(p^2). A participant initially struggles to solve for λ but later confirms they found the solution after receiving guidance. The final expression for Pr(M>1) is derived as 1 - Pr(M=0) - Pr(M=1), which simplifies to 1 - e^(-λ) - λe^(-λ). The thread concludes with the participant successfully resolving their query.
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The problem is the following;

N has a geometric distribution with Pr(N=0)>0. M has a Poisson distribution. You are given:

E(N) = E(M); Var(N) = 2Var(M)

Calculate Pr (M>1).

From general knowledge we know that the expected value of a variable in a geometric distribution E(N) = q/p, and Var(N) = q/(p^2).
Also; the expected value of a variable in a Poisson distribution E(M) = lambda and Var(M) also = lambda.

I believe that the answer is 1 - pr(M=0) - pr(M=1) which is the equivalent of
1-e^(-lambda)-lambda*e^(-lambda).

But this would require solving for lambda, a feat I have not yet accomplished.


Any pointers?..

Thanks in advance,

Teddy
 
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You should be able to solve for p and \lambda, from \lambda = (1-p)/p, and 2\lambda = (1-p)/(p^2).

Note that Pr(N=0) = p > 0 so \lambda < +\infty.
 
Last edited:
Thanks Enuma, I was able to solve for it.
 
NM i saw the reply.
 
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