Poisson Distribution

The Poisson distribution with parameter l is the discrete probability distribution of a random variable X with the following probability mass function:

The Poisson distribution is often used to model the probability distribution of the number of outcomes that are labeled 'successes' during a given time interval or within a specified region. The time interval involved could have a variety of lengths, e.g., a second, minute, hour, day, year, and multiples thereof. The region in question could be a line segment, an area, a volume, or some n-dimensional space, where n is an integer. Experiments of this type are known as Poisson experiments, and they possess the following characteristics:

1. The number of outcomes occurring in any given time interval or region is independent of the number of outcomes occurring in any other disjoint time interval or region.
2. The probability of a single outcome occurring in a very short time interval or very small region is proportional to the length of the time interval or the size of the region. This value is not affected by the number of outcomes occurring outside this particular time interval or region.
3. The probability of having more than one outcome occurring in a very short time interval or very small region is negligible.

The mean and variance of a random variable X that has a Poisson distribution with parameter l are both equal to each other and to the parameter l, i.e.,

E(X) = Var(X) = l

EX. In one particular autobiography of a professional athlete, there are an average of 15 spelling errors per page. If the Poisson distribution is used to model the probability distribution of the number of errors per page, then the random variable X, the number of errors per page, has a Poisson distribution with l = 15. The probability that there are no errors on a page is e^-1515⁰ = 3.06 x 10^-7.

(How much would you pay for a book with such a lack of spelling ability?)