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A routine exercise designed to test one's knowledge; often from a textbook, course, or test used for a class or self-study. This community's policy is to "provide helpful hints" for such questions rather than complete answers.
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Conditional probability of sum of $n$ Poisson variables given sum of subset of $n$ variables...
I have $n$ Poisson random variables $X_i$ each sampled from an independent Poisson distribution parametrized by $\lambda_i$. I want to compute $P(X_1+X_2+...+X_n=z|X_1+X_3=y)$. The distribution of the …
3
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Conditional probability of sum of $n$ Poisson variables given sum of subset of $n$ variables...
The sum of independent Poisson random variables is a Poisson random variable as shown here. So for $Z = X+Y$ where all are Poisson r.v., $P(Z=z)=\frac{\exp^{-\lambda}\lambda^z}{z!}$. Here, $\lambda=\l …