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Approximating the poisson

  1. May 23, 2013 #1
    1. The problem statement, all variables and given/known data

    During a two hour window, people are given the option of calling number X, donating $9.90, or number Y, donating $0.50.

    X is Poisson distributed with 1500 calls/minute. Y is Poisson with 3750 calls/minute.

    What is the probability that more than $2,000,000 is raised?

    2. Relevant equations

    3. The attempt at a solution

    X = number of calls to number X in 120 minutes = Po(120*1500) = Po(180000)

    Y = number of calls to number Y in 120 minutes = Po(120*3750) = Po(450000)

    Let Z = 9.9X + 0.5Y. We're looking for P(Z > 2,000,000).

    Z should have an expectation of 1,782,000 + 225,000 = 2,007,000.

    Now I want to approximate Z = Po(μ) with Z ≈ N(μ,σ). But I can't work out the standard deviation for Z. Using [itex]\sqrt{μ}[/itex] doesn't give me the right answer.
  2. jcsd
  3. May 23, 2013 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    I would bet that you have already seen how to do it, but may have forgotten. So, if X and Y are independent random variables and a, b are numbers, how does the variance of Z = a*X + b*Y relate to Var(X) and Var(Y)? There is a standard formula. It is used over and over and over again, so you should get to know it if you don't already or have forgotten it.
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