Calculating the probability of a sum of Poisson-distributed random variables

Question

In Java, I have a set of users, each of which has a Poisson-distributed demand with a known mean:

import org.apache.commons.math3.distribution.PoissonDistribution;

public class User {
  private int mean;
  private PoissonDistribution dist;
  public void setMean(int mean) {
      this.mean = mean;
      this.poissonDistribution = new PoissonDistribution(mean);
  }
}

...
User u1 = new User(); u1.setMean(20);
User u2 = new User(); u2.setMean(30);
User u3 = new User(); u3.setMean(40);

Now I'd like to calculate the probability that all Users have a cumulated demand below a certain value:

double probabilityBelowX = calculateCumulatedProbability(50, u1, u2, u3); // <- what must this method look like?

I am stuck at the question, how to solve this problem in Java. Am I missing something in the math package? I know that the demands of the customers are independent, so according to my knowledge about the Poisson distribution I can simply add up the values. But I only have cumulativeProbability(int x) for every single one of the user demands, but not for several at once.

Answer 1

If you assume that customers demands are independant and Poisson, the total demand will be a Poisson distribution as well with parameter equal to the sum of the individual Poisson parameters (see wikipedia for example).

A C++ code example (I am not so familiar with Java):

double cumulative(double x, User * users, int nusers) 
{
    double lambda = 0.;
    for (unsigned i = 0; i < nusers; ++i) lambda += users[i].mean();
    return PoissonDistribution(lambda).cumulativeProbability(x);
}