Loss Models: From Data to Decisions, 4th Edition

6.2 The Poisson distribution

The pf for the Poisson distribution is

The probability generating function from Example 3.8 is

The mean and variance can be computed from the probability generating function as follows:

For the Poisson distribution, the variance is equal to the mean. The Poisson distribution and Poisson processes (which give rise to Poisson distributions) are discussed in many textbooks on probability, statistics and actuarial science, including Panjer and Willmot [89] and Ross [98].

The Poisson distribution has at least two additional useful properties. The first is given in the following theorem.

Theorem 6.1 Let N₁, …, N_n be independent Poisson variables with parameters Λ₁, …, Λ_r Then N = N₁ + ··· + N_n has a Poisson distribution with parameter Λ₁ + ··· + Λ_n.

Proof: The pgf of the sum of independent random variables is the product of the individual pgfs. For the sum of Poisson random variables we have

where Λ = Λ₁ + ··· + Λ_n. Just as is true with moment generating functions, the pgf is unique and, therefore, N must have a Poisson distribution with parameter Λ.

The second ...