# Approximating the poisson

1. May 23, 2013

### Gauss M.D.

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 $\sqrt{μ}$ doesn't give me the right answer.

2. May 23, 2013

### Ray Vickson

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.