Spread of total daily insurance claims (random sum)

Question

Claims arrive at an insurer as a Poisson count $N$ averaging 10 per day. Each claim's size $X$ is independent with mean $500$ dollars and standard deviation $300$ dollars, and claim sizes are independent of how many claims there are. Let $S$ be the total amount paid out in a day.

What are the mean and variance of $S$ ?

Accepted Answer

This is a random sum: a random number of terms, each a random size. Two layers of randomness feed the spread, and the law of total variance combines them. Mean is the easy part, expected count times expected size: \mathbb{E}[S] = \mathbb{E}[N]\,\mathbb{E}[X] = 10 	imes 500 = 5000. Variance has two contributions: \operatorname{Var}(S) = \underbrace{\mathbb{E}[N]\operatorname{Var}(X)}{	ext{sizes vary}} + \underbrace{\operatorname{Var}(N)\,\mathbb{E}[X]^2}{	ext{count varies}}. The first term is the spread from claim sizes wobbling; the second is the spread from the number of claims wobbling, …

Spread of total daily insurance claims (random sum)

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