# Poisson process - expected value of arrivals

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• Nov 27th 2017, 05:19 PM
fatty
Poisson process - expected value of arrivals
Hello. I want to know if I am on the right track when solving this problem:

"Assume that customers arrive at a bank in accordance with a Poisson process with rate λ = 6 per hour, and suppose that each customer is a man with probability 2/3 and a woman with probability with 1/3. Now suppose that 10 men arrived in the first 2 hours. How many women would you expect to have arrived in the first 2 hours?"

This kind of problem is relatively new to me, so forgive me if my questions seem ridiculous.

Let's say that A is the event of the number of men arriving in two hours and B are the corresponding events for women.
Are these two events independent of each other?

Is it correct to say that the expected value of total arrivals in two hours are: λ * t = 12 arrivals? Or have I misinterpreted the meaning of λ?
If E[X(2)]=12 is correct so E[women arrival in 2 hours]= 12*(1/3) = 4.

I have a feeling that this was way to easy to solve. Hope some of you guys can give me a hand with this.
• Nov 28th 2017, 03:33 AM
HallsofIvy
Re: Poisson process - expected value of arrivals
Quote:

Is it correct to say that the expected value of total arrivals in two hours are: λ * t = 12 arrivals? Or have I misinterpreted the meaning of λ?
If E[X(2)]=12 is correct so E[women arrival in 2 hours]= 12*(1/3) = 4.
What if you applied this same reasoning to the number of men? If "the expected value of total arrivals in two hours are: λ * t = 12 arrivals" and 2/3 of the arrivals are men, then number of men should be (2/3)(12)= 8 but we are told that "10 men arrived in the first 2 hours".

It looks to me like the "Poisson distribution" information is red herring. You are told that "the probability that a customer is a man is 2/3 and the probability that a customer is a woman is 1/3". If 10 men have arrived, no matter what the rate at which they arrived, then we would expect that 5 women have arrived.