保険金支払いをモデル化する
保険会社によって支払われた保険金の平均を，損失分布の点別の変換を使って，計算する．
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[{{"損失の確率 = ", q}, {"損失分布 = ", ExponentialDistribution[a]}, {"控除免責金額 = ", d}, {"給付金の限度額 = ", s}, {"保険金支払いの平均 = ", m}}, BaseStyle -> {FontFamily -> "Verdana"}, Alignment -> {{Right, Left}, Automatic}], FontSize -> 14, Bold]]