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The mall has 3 entrances A, B, C and is open 10 hours a day. The number of people entering the mall at any hour through the entrances A, B, C is Poisson (20), Poisson (30), Poisson (50) respectively. The number of people entering each entrance is independent of the other entrances and independent of different hours. The Question: It is known that 50 people entered the mall in the first hour (from all entrances together). What is the expectation of the number of people entering from Entrance B in the first hour?

I tried to break down to indicators (50 indicators each indicator represents whether or not a person entered through B) and it did not work. I also tried the complete expectation formula but it seemed too complicated.

The final answer is 15.

StubbornAtom
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Ella
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1 Answers1

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Conditional on the total number of events, the events are distributed over the contributing Poisson processes in proportion to their rates; see e.g. Prove that $X|X + Y$ is a Binomial random variable and Poisson random variables and Binomial Theorem. The proportion of $B$ is $\frac{30}{20+30+50}=\frac3{10}$, so the expected number of events from that process is $\frac3{10}\cdot50=15$.

joriki
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