Expected donations during a broadcast

Question

During a live fundraiser, donations arrive as a Poisson process at rate 12 per hour. Independently, each donation averages \8. The broadcast runs 3 hours. What is the expected total amount donated?

Accepted Answer

The total raised is a compound Poisson sum, and its expectation is the expected number of donations times the average size of each. The expected number of donations is E[N] = \lambda t = 12 	imes 3 = 36. Each averages \8, so E[	ext{total}] = E[N] 	imes E[	ext{gift}] = 36 	imes 8 = \288. In words: about 36 donations are expected, each worth about \8, for roughly \288. Takeaway. The expected running total is always (expected number of contributions) times (average contribution), no matter how the individual amounts vary.

Expected donations during a broadcast

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