Mathematics · Probability

Let X 0 = 0 and for k ≥ 1 let X k be a random variable with Binomial(k, 1 2 ) distribution. Let N be a Poisson random variable with mean 1. Assume that for every k ≥ 1, X k and N are independent and set Y = X N . Given that Y = 3, what is the probability that N = 3?

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e -1/2

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Let X 0 = 0 and for k &ge; 1 let X k be a random variable with Binomial(k, 1 2 ) distribution. Let N be a Poisson random variable with mean 1. Assume that for every k &ge; 1, X k and N are independent and set Y = X N . Given that Y = 3, what is the probability that N = 3?