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A discrete, integer-valued random variable X is said to be Poisson distributed with parameter L if for non-negative k

P(X = k) = e-LLk/k!

The distribution has mean and variance L. The sum of two Poisson distributed variables with parameters L, M is itself Poisson distributed and has parameter L + M.

It turns out that the Poisson distribution is one of the distributions that tend to pop up naturally. What is the explanation for this?

Proposition:
Let Xn be a sequence of random variables, each binomially distibuted with parameters n, p, with L = np is constant. Then P(Xn = k) → e-LLk/k! as n → ∞.

Proof:
Using the definition of the binomial distribution we have
P(Xn = k) = (n!/k!(n-k)!)pk(1-p)n-k = (n!/(n-k)!nk)((np)k/k!)(1-L/n)n(1-L/n)-k → e-LLk/k!
as n → ∞.

So the Poisson distribution can be considered as a limiting case of the binomial distribution. A few examples should illustrate the importance of this.

The distribution of misprints on the pages of a book can be modelled by saying that there are n = 1000 characters on each page, and each of these have an independent probablity of say p = 0,0001 of being a misprint. n is large, p is small but L = np = 0,1 is moderate in size. Therefore the Poisson distribution with parameter L = 0,1 will provide a good approximation to the number of misprints on each page.

Suppose that a radioactive sample has an average of L decays in an interval of time. To approximate the distribution of the number of decays in the interval we divide it into a large number n subintervals. For large n the probability probability of more than one decay in a subinterval is negligable. The probability that there is a decay in an interval is p = L/n. As n increases the approximation becomes more accurate, and since np = L remains constant the limiting case is the Poisson distribution with parameter L.

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