Model Claim Payments for Insurance
Compute the mean claim payment made by an insurance company, using a piecewise transformation of the loss distribution.
LossCDF = ((1 - q) + q (1 - E ^ (-a x))) HeavisideTheta[x];Loss𝒟 = ProbabilityDistribution[{"CDF", LossCDF}, {x, -∞, ∞}, Assumptions -> a > 0 && 0 < q < 1];
PayoutFunction[x_] := Piecewise[{{0, x ≤ d}, {x - d, d < x < s + d}, {s, x ≥ s + d}}];g = Plot[PayoutFunction[x] /. {d -> 400, s -> 1000}, {x, 0, 1800}, PlotStyle -> {Thickness[0.01]}, Filling -> Axis, Exclusions -> None];Payout𝒟 = TransformedDistribution[PayoutFunction[x], xLoss𝒟, Assumptions -> d > 0 && s > 0 && 0 < q < 1 && a > 0];m = Mean[Payout𝒟];Show[g, ImageSize -> 500, PlotLabel -> Style[Grid[{{"Loss Probability = ", q}, {"Loss Distribution = ", ExponentialDistribution[a]}, {"Deductible = ", d}, {"Benefit Limit = ", s}, {"Mean Claim Payment = ", m}}, BaseStyle -> {FontFamily -> "Verdana"}, Alignment -> {{Right, Left}, Automatic}], FontSize -> 14, Bold]]