Define a compound Poisson distribution:

Mean and variance:

Simulate a compound Poisson distribution:

Simulate a compound Poisson distribution for different renewal rates:

Distribution parameter estimation:

Estimate the distribution parameters from sample data:

Compare the histogram of the data with a sample from the estimated distribution:

Define a compound Poisson distribution:

Skewness:

Kurtosis:

Define a compound Poisson distribution:

Calculate cumulants:

Compare to the reward distribution:

Compare to the renewal distribution:

Different moments with closed forms as functions of parameters:

Moment:

CentralMoment:

FactorialMoment:

Cumulant:

Generating functions for a CompoundPoissonDistribution:

Characteristic function:

Moment generating function:

Factorial moment generating function:

Cumulant generating function:

Compound Poisson distribution with a mixture jump distribution:

Mean of the distribution:

Compare with the value obtained from simulation:

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 Weibull distribution with shape parameter 10 and scale parameter 3. Find the expected total cost of the gifts given by the store during the 12-hour period for which the store is open on that day:

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

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

Aggregate claims from a risk have a compound Poisson distribution with Poisson parameter 200 and an individual claim amount distribution, which 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. Calculate the mean and variance of the reinsurer's aggregate claims. The individual claims for the reinsurer follow a transformed Pareto distribution, since the reinsurer only pays amounts greater than $300:

The aggregate claims:

Mean and variance of the reinsurer's aggregate claims:

CompoundPoissonDistribution is a slice distribution of CompoundPoissonProcess:

CompoundPoissonDistribution with BernoulliDistribution is a PoissonDistribution:

CompoundPoissonDistribution with a special BorelTannerDistribution follows PoissonConsulDistribution:

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

Compare characteristic functions:

Renewal rate influences the kurtosis:

Use a different reward distribution:

Slice of CompoundRenewalProcess with renewal time given by ExponentialDistribution:

Create a random sample for rate 3 and slice at 7:

Compare to a sample of the corresponding CompoundPoissonDistribution:

The default parameter estimation using maximum likelihood might take a long time:

Use a different method: