CumulantGeneratingFunction
CumulantGeneratingFunction
CumulantGeneratingFunction[dist,t]
给出分布 dist 的累积量母函数，函数的自变量为 t.
CumulantGeneratingFunction[dist,{t1,t2,…}]
给出多元分布 dist 的累积量母函数，函数的自变量为 t1、t2、….
更多信息
- CumulantGeneratingFunction[dist,t] 由 Log[MomentGeneratingFunction[dist,t]] 得到.
- CumulantGeneratingFunction[dist, {t1,t2,…}] 由Log[MomentGeneratingFunction[dist,{t1,t2,…}]] 得到.
- i
阶累积量可以通过SeriesCoefficient[cgf,{t,0,i}]i! 从累积量母函数 cgf 中提取得到.
范例
打开所有单元 关闭所有单元基本范例 (3)
范围 (5)
CumulantGeneratingFunction[ProbabilityDistribution[(3/4)Sqrt[(1 + x/2)], {x, -1, 1}], t]CumulantGeneratingFunction[TransformedDistribution[u z, {uUniformDistribution[], zNormalDistribution[]}], t]hdist = HistogramDistribution[ExampleData[{"Statistics", "FatigueLifeFailures"}]]CumulantGeneratingFunction[hdist, t]CumulantGeneratingFunction[TruncatedDistribution[{-10, 10}, CauchyDistribution[0, 1]], t]CumulantGeneratingFunction[PoissonProcess[μ][s], t]应用 (5)
两个独立随机变量之差的累积量母函数等于具有相反符号参数的累积量母函数之和：
TransformedDistribution[x - y, {xPoissonDistribution[μ], yPoissonDistribution[λ]}]CumulantGeneratingFunction[SkellamDistribution[μ, λ], t]% == CumulantGeneratingFunction[PoissonDistribution[μ], t] + CumulantGeneratingFunction[PoissonDistribution[λ], -t]dist = LaplaceDistribution[μ, σ];cgf[t_] = CumulantGeneratingFunction[TransformedDistribution[(x - Mean[dist]) / StandardDeviation[dist], xdist], t]
n * cgf[t / Sqrt[n]]
Limit[n * cgf[t / Sqrt[n]], n -> Infinity]CumulantGeneratingFunction[NormalDistribution[], t]对服从 GammaDistribution 的损失进行的投保，求 Esscher 保险费：
D[CumulantGeneratingFunction[GammaDistribution[α, β], h], h](Expectation[x Exp[h x], xGammaDistribution[α, β]]/Expectation[Exp[h x], xGammaDistribution[α, β]])构建生存函数的 Bernstein–Chernoff 界限 
：
dist = BinomialDistribution[n, 1 / 2];
ℐ[t_, x_] = t * x - CumulantGeneratingFunction[dist, t];
k = Mean[dist] + u StandardDeviation[dist];
assumps = t > 0∧n > 0∧k < n∧u > 0;{bound} = Simplify[Exp[-ℐ[t, k]] /. Solve[D[ℐ[t, k], t] == 0 && assumps, t, Reals], assumps]Table[LogPlot[{SurvivalFunction[dist, k], bound}, {u, 0, 5}, PlotPoints -> n], {n, {50, 100, 500, 1000}}]
Series[bound, {n, ∞, 2}]//Simplify构建 VarianceGammaDistribution 的 PDF 的丹尼尔鞍点逼近：
vgd = VarianceGammaDistribution[7 / 3, 3, -1, 0];
cgf[t_] = CumulantGeneratingFunction[vgd, t]
sol = Solve[cgf'[t] == x, t]
Limit[t /. sol, x -> 0]ts = Last[t /. sol]approxPDF[x_] = (1/Sqrt[2Pi cgf''[ts]])Exp[cgf[ts] - ts x]//FullSimplifynorm = NIntegrate[approxPDF[x], {x, -Infinity, Infinity}]Plot[{approxPDF[x] / norm, PDF[vgd, x]}, {x, -4, 4}, PlotLegends -> {"Saddlepoint approximation", "Exact density"}]属性和关系 (3)
CumulantGeneratingFunction 的指数即 MomentGeneratingFunction：
CumulantGeneratingFunction[NormalDistribution[μ, σ], t]Exp[%] == MomentGeneratingFunction[NormalDistribution[μ, σ], t]CumulantGeneratingFunction 是累积量序列的指数母函数：
Cumulant[PoissonDistribution[μ], r]ExponentialGeneratingFunction[%, r, t]直接使用 CumulantGeneratingFunction：
CumulantGeneratingFunction[PoissonDistribution[μ], t]
Cumulant[UniformDistribution[{0, 1}], 4]Limit[D[CumulantGeneratingFunction[UniformDistribution[{0, 1}], t], {t, 4}], t -> 0]使用 SeriesCoefficient 的公式化形式：
With[{r = 4}, SeriesCoefficient[CumulantGeneratingFunction[UniformDistribution[{0, 1}], t], {t, 0, r}]r!]可能存在的问题 (2)
Table[Cumulant[ParetoDistribution[1, 4], r], {r, 5}]相应地，CumulantGeneratingFunction 未定义：
CumulantGeneratingFunction[ParetoDistribution[1, 4], t]CumulantGeneratingFunction 的解析式可能未知：
CumulantGeneratingFunction[LogNormalDistribution[μ, σ], t]直接使用 Cumulant 求累积量：
Cumulant[LogNormalDistribution[μ, σ], 3]巧妙范例 (1)
dists = {NegativeBinomialDistribution[10, 1 / 3], PoissonDistribution[3], BorelTannerDistribution[5 / 6, 10], ExponentialDistribution[1], BirnbaumSaundersDistribution[1, 1 / 3], HyperbolicDistribution[2, 1, 1, 2]};Table[Plot3D[Re[CumulantGeneratingFunction[𝒟, x + I y]]//Evaluate, {x, -2, 2}, {y, -4, 4}, Mesh -> None, ImageSize -> 200, PlotLabel -> 𝒟], {𝒟, dists}]
Wolfram Research (2010)，CumulantGeneratingFunction，Wolfram 语言函数，https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html.
文本
Wolfram Research (2010)，CumulantGeneratingFunction，Wolfram 语言函数，https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html.
CMS
Wolfram 语言. 2010. "CumulantGeneratingFunction." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html.
APA
Wolfram 语言. (2010). CumulantGeneratingFunction. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html 年