We present a discrete example of a compound Poisson distribution. A random variable has a compound distribution if where the number of terms is a discrete random variable whose support is the set of all nonnegative integers (or some appropriate subset) and the random variables are identically distributed (let be the common distribution). We further assume that the random variables are independent and each is independent of . When follows the Poisson distribution, is said to have a compound Poisson distribution. When the common distribution for the is continuous, is a mixed distribution if is nonzero. When the common distribution for the is discrete, is a discrete distribution. In this post we present an example of a compound Poisson distribution where the common distribution is discrete. The compound distribution has a natural insurance interpretation (see the following links).
In general, the distribution function of a compound Poisson random variable is the weighted average of all the convolutions of the common distribution function of the individual claim amount . The following shows the form of such a distribution function:
where is the common distribution of the and is the convolution of .
If the distribution of the individual claim is discrete, we can obtain the probability mass function of by convolutions as follows:
Suppose the number of claims generated by a portfolio of insurance policies over a fixed time period has a Poisson distribution with parameter . Individual claim amounts will be 1 or 2 with probabilities 0.6 and 0.4, respectively. For the compound Poisson aggregate claims , find for .
The probability mass function of is: where . The individual claim amounnt has a Bernoulli distribution since it is a two-valued discrete random variable. For convenience, we let (i.e. we consider is a success). Then the sum has a Binomial distribution. Consequently, the convolution is simply the distribution function of Binomial(n,p). The following shows for .
Since we are interested in finding for , we only need to consider . The following matrix shows the relevant values of . The rows are for . The columns are , , , , .
To obtain the probability mass function of , we simply multiply each row by where .