Spread of total monthly rainfall (random sum)

Question

Storms hit a town as a Poisson count N averaging 3 per month. Each storm drops rainfall X that is independent with mean 2 inches and standard deviation 1 inch, independent of the number of storms. Let S be the total monthly rainfall. What are the mean and variance of S?

Accepted Answer

This is a random sum (random number of storms, each a random amount). The mean is expected count times expected amount: \mathbb{E}[S] = \mathbb{E}[N]\,\mathbb{E}[X] = 3 	imes 2 = 6 	ext{ inches}. The variance combines both layers via the law of total variance: \operatorname{Var}(S) = \mathbb{E}[N]\operatorname{Var}(X) + \operatorname{Var}(N)\,\mathbb{E}[X]^2. For a Poisson count, \operatorname{Var}(N) = 3. With \operatorname{Var}(X) = 1^2 = 1 and \mathbb{E}[X]^2 = 2^2 = 4: \operatorname{Var}(S) = 3(1) + 3(4) = 3 + 12 = 15. Standard deviation \sqrt{15} \approx 3.87 inches. Most of the varian…

Spread of total monthly rainfall (random sum)

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