Simulate a compound Poisson process with an exponential jump size distribution:

Mean and variance functions:

Shoppers arrive at a newly renovated store according to a Poisson process with a rate of 20 customers per hour. The store promotes this event by giving every customer a gift. The gift has a value that follows a WeibullDistribution with shape parameter 10 and scale parameter 3. Simulate the cost of the gift process during the 12-hour period for which the store is open that day and find the expected total cost to the store:

Simulate the process for 12 hours:

Expected total cost of the gifts given on the inaugural day:

Simulate gift cost distribution:

Probability density function of cost distribution:

Empirical probability that the store spends between $500 and $800 on the gifts:

Aggregate claims from a risk follow a compound Poisson process with Poisson parameter 200. The claim amount distribution is a Pareto distribution with minimum value parameter 300, shape parameter 3, and location parameter 0. The insurer has effected excess of loss reinsurance with retention level 300. Simulate the claims process for four years:

Slice distributions for the first four years:

Mean and variance of the reinsurer's aggregate claims for the first four years:

CompoundPoissonProcess is a jump process:

The renewal rate λ controls the frequency of the jumps:

Compound Poisson process is not weakly stationary:

Any slice distribution of Compound Poisson process is a CompoundPoissonDistribution:

Slice of compound Poisson process with BernoulliDistribution follows PoissonDistribution:

Slice of compound Poisson process with a special BorelTannerDistribution follows PoissonConsulDistribution:

The sum of Borel-Tanner distributed variables follows Borel-Tanner distribution, hence the slice distribution is equivalent to the parameter mixture distribution::

Compare characteristic functions: