Spread of hourly website revenue (random sum)

Question

Buyers arrive at a website as a Poisson count $N$ averaging 20 per hour. Each order value $X$ is independent with mean $40$ dollars and standard deviation $30$ dollars, independent of the number of orders. Let $S$ be the total revenue in an hour.

What are the mean and variance of $S$ ?

Accepted Answer

This is a random sum (random number of orders, each a random value). The mean is expected count times expected value: \mathbb{E}[S] = \mathbb{E}[N]\,\mathbb{E}[X] = 20 	imes 40 = 800. The variance combines two sources through 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) = \mathbb{E}[N] = 20. With \operatorname{Var}(X) = 30^2 = 900 and \mathbb{E}[X]^2 = 40^2 = 1600: \operatorname{Var}(S) = 20(900) + 20(1600) = 18000 + 32000 = 50000. Standard deviation \sqrt{50000} \app…

Spread of hourly website revenue (random sum)

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