Poisson Approximation to the Binomial Distribution

The random variable X that is binomially distributed with parameters p and n has the following probability mass function:

The mean and variance of X are

E(X) = np
Var(X) = np(1-p)

If the value of n is large, and the value of p is close to zero, then the binomial distribution with parameters p and n can be approximated by a Poisson distribution with parameter l = np. A random variable X having this Poisson distribution will have a mean and variance whose value is l = np, i.e.,

E(X) = Var(X) = l = np

These values are the same as the mean and variance of a binomial distribution with parameters p and n, except that the values of the variances differ by a factor of (1-p). However, if the value of p is close to zero, the value of (1-p) is close to 1.

EX. A certain rare disease affects 1 out of every 10,000 persons in the US. The random variable X, the number of people in a sample of 100,000 that are afflicted with this disease, is binomially distributed with n = 100,000 and p = 0.0001. Since n is very large and p is close to zero, the Poisson approximation to the binomial distribution should provide an accurate estimate. Thus, the distribution of X approximates a Poisson distribution with l = np = (100000)(0.0001) = 10.

The probability that there are no more than 2 persons in the sample of 100,000 that have the misfortune of contracting the disease is