September 2012
Beginner to intermediate
536 pages
14h 40m
English
book
Loss Models: From Data to Decisions, 4th Edition
by Stuart A. Klugman, Harry H. Panjer, Gordon E. Willmot
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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, …, Nn be independent Poisson variables with parameters Λ1, …, Λr Then N = N1 + ··· + Nn has a Poisson distribution with parameter Λ1 + ··· + Λn.
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 + ··· + Λn. 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 ...
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