Model Aggregated Claims Value with Compound Poisson Distribution 

Values of claims of certain collision insurance with a $500 deductible follow a ParetoDistribution. The number of claims per year follows a PoissonDistribution and averages 20 claims per month. Hence the aggregated claims value follows a CompoundPoissonDistribution.

In[1]:=

X \[ScriptCapitalD] = CompoundPoissonDistribution[12*20, ParetoDistribution[500, 3, 0]];

The average value of all the claims per year.

In[2]:=

X Mean[\[ScriptCapitalD]]

Out[2]=

Find the probability density function of the aggregated claims value using simulation.

In[3]:=

X simulation = RandomVariate[\[ScriptCapitalD], 10^5]; Histogram[simulation, {40000, 120000, 1000}, "PDF", ImageSize -> 300]

Out[3]=

Use the distribution obtained from the simulation to compute the probability that the aggregated claims have a value higher than 80,000.

In[4]:=

X Probability[x > 80000, x \[Distributed] simulation] // N

Out[4]=