Mathematics · Probability

Let X 1 , X 2 , ... be i.i.d. Normal random variables with mean 2 and variance 3. Let N be a Poisson random variable with mean 4 that is independent of {X 1 , X 2 , ...}. Let Y = X 1 + ... + X N if N ≥ 1, and Y = 0 if N = 0. What is the variance of Y?

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Let X 1 , X 2 , ... be i.i.d. Normal random variables with mean 2 and variance 3. Let N be a Poisson random variable with mean 4 that is independent of {X 1 , X 2 , ...}. Let Y = X 1 + ... + X N if N ≥ 1, and Y = 0 if N = 0. What is the variance of Y?

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Let X 1 , X 2 , ... be i.i.d. Normal random variables with mean 2 and variance 3. Let N be a Poisson random variable with mean 4 that is independent of {X 1 , X 2 , ...}. Let Y = X 1 + ... + X N if N &ge; 1, and Y = 0 if N = 0. What is the variance of Y?

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Let X 1 , X 2 , ... be i.i.d. Normal random variables with mean 2 and variance 3. Let N be a Poisson random variable with mean 4 that is independent of {X 1 , X 2 , ...}. Let Y = X 1 + ... + X N if N ≥ 1, and Y = 0 if N = 0. What is the variance of Y?