Binomial Approximation to a Poisson Random Variable

sample size n

20

λ

10

If a coin that comes up heads with probability

p

is tossed

n

times, the number of heads observed follows a binomial probability distribution. The Poisson probability distribution can be regarded as a limiting case of the binomial distribution as the number of tosses grows and the probability of heads on a given toss is adjusted to keep the expected number of heads constant. More precisely, if

X

n

is a binomial random variable with parameters

n

and

λ/n

, then the probability distribution for

X

n

approaches a Poisson distribution with parameter λ as

n

approaches infinity.

The probability histogram for the binomial random variable is blue and the one for the Poisson random variable is red. A measure of agreement between the two is obtained by computing the purple area—100% would represent complete agreement between the two distributions.