Claims in a quiet or busy month, each with a cost

Question

An insurer's month is quiet or busy, each with probability $\tfrac12$ . A quiet month brings a random number of claims averaging $2$ ; a busy month brings a random number averaging $6$ . Each claim, whenever it happens, costs an amount averaging $500, independently of everything else.

What total monthly payout should you expect?

Accepted Answer

Two layers of randomness feed a third: the month type sets the average number of claims, the number of claims is random, and each claim's cost is random. Peel them with the tower rule. Step 1, expected number of claims. The claim count N depends on the month type, so average the two month types (tower rule): E[N] = 	frac12 	imes 2 + 	frac12 	imes 6 = 1 + 3 = 4. Step 2, total payout as a random-length sum. The payout adds up N claim costs, each averaging \500. Conditioning on N, the expected payout given N = n is 500\,n, so E[	ext{payout}] = E[500 \, N] = 500 \, E[N] = 500 	imes 4 = 2000…

Claims in a quiet or busy month, each with a cost

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