FactorialMomentGeneratingFunction
FactorialMomentGeneratingFunction
FactorialMomentGeneratingFunction[dist,t]
给出分布 dist 的阶乘矩母函数，函数的自变量为 t.
FactorialMomentGeneratingFunction[dist,{t1,t2,…}]
给出多元分布 dist 的阶乘矩母函数，函数的自变量为 t1、t2、….
更多信息
- FactorialMomentGeneratingFunction 也称为概率母函数（pgf）.
- FactorialMomentGeneratingFunction[dist,t] 等价于 Expectation[tx,xdist].
- FactorialMomentGeneratingFunction[dist, {t1,t2,…}] 等价于 Expectation[t1x1t2x2…,{x1,x2,…}dist].
- i
阶阶乘矩可以通过 SeriesCoefficient[fmgf,{t,1,i}]i! 从阶乘矩母函数 fmgf 中提取得到. - 一个离散随机变量取值为 i 的概率可以通过 SeriesCoefficient[expr,{t,0,i}] 从阶乘矩生成函数 expr 中提取得到.
范例
打开所有单元 关闭所有单元基本范例 (3)
FactorialMomentGeneratingFunction[PoissonDistribution[μ], t]FactorialMomentGeneratingFunction[NormalDistribution[μ, σ], t]FactorialMomentGeneratingFunction[MultinomialDistribution[n, {p1, p2, p3}], {t1, t2, t3}]范围 (5)
FactorialMomentGeneratingFunction[ProbabilityDistribution[(18^-k/Sqrt[6]) Binomial[4k, 2k], {k, 0, Infinity, 1}], t]FactorialMomentGeneratingFunction[EmpiricalDistribution[{1, 2, 3, 3, 4, 5, 5}], t]FactorialMomentGeneratingFunction[CensoredDistribution[{0, 10}, PoissonDistribution[2]], t]FactorialMomentGeneratingFunction[ParameterMixtureDistribution[PoissonDistribution[λ], λExponentialDistribution[μ]], t]FactorialMomentGeneratingFunction[PoissonProcess[μ][s], t]应用 (6)
FactorialMomentGeneratingFunction[GeometricDistribution[p], t] ^ n与 NegativeBinomialDistribution 的 fmgf 相比较：
FactorialMomentGeneratingFunction[NegativeBinomialDistribution[n, p], t]% - %%求 
个独立同分布的几何分布变量的和的 fmgf，其中随机数 
服从 PoissonDistribution：
Expectation[FactorialMomentGeneratingFunction[GeometricDistribution[p], t] ^ n, nPoissonDistribution[μ]]与 PolyaAeppliDistribution 的 fmgf 相比较：
FactorialMomentGeneratingFunction[PolyaAeppliDistribution[μ(1 - p), 1 - p], t]Simplify[% / %%]从一个非负整数随机变量的 fmgf 求它的 PDF：
fmgf[t_] = ((2 - t)^3/(4 - 3 t)^5);pdf = SeriesCoefficient[fmgf[z], {z, 0, k}, Assumptions -> k ≥ 0]DiscretePlot[pdf, {k, 0, 25}, FillingStyle -> Gray]Sum[pdf, {k, 0, Infinity}]对 BernoulliDistribution 建立一个概率母函数：
bpgf[z_] = FactorialMomentGeneratingFunction[BernoulliDistribution[p], z]{pgf[z_]} = u /. Solve[u == z bpgf[u], u] //Simplify与进行平移后的 GeometricDistribution 的概率母函数相比较：
FactorialMomentGeneratingFunction[TransformedDistribution[x + 1, xGeometricDistribution[1 - p]], z]对 GeometricDistribution 的概率母函数（pgf）应用拉格朗日变换：
gpgf[z_] = FactorialMomentGeneratingFunction[GeometricDistribution[p], z]u /. Solve[u == z gpgf[u] && 0 < p < 1 && 0 < u < 1 && 0 < z < 1, u, Reals]{hpgf[z_]} = Refine[%, 0 < p < 1 && 0 < z < 1]重建 PDF：
pdf = SeriesCoefficient[hpgf[z], {z, 0, k}, Assumptions -> 0 < p < 1]//FullSimplify
Sum[pdf, {k, 1, Infinity}, Assumptions -> 0 < p < 1]DiscretePlot[pdf /. p -> 5 / 9, {k, 0, 10}]对正反面出现概率不同的一个硬币进行投掷，求若要连续两次出现正面，所需投掷次数的分布. 令 
为正面出现的概率. 事件空间由三种事件类型组成：反面（T）、先正面后反面（HT），连续出现两次正面（HH）. 其概率分别是：
pr["T"] = (1 - p);
pr["HT"] = p(1 - p);
pr["HH"] = p ^ 2;求感兴趣的随机变量的 fmgf，把它解释为 T 事件的总数加上 HT 事件的总数加上2：
fmgf = t ^ 2FactorialMomentGeneratingFunction[NegativeMultinomialDistribution[1, {pr["T"], pr["HT"]}], {t, t^2}]重建 PDF：
pdf = SeriesCoefficient[fmgf, {t, 0, k}, Assumptions -> k ≥ 0]mean = Limit[D[fmgf, t], t -> 1]//SimplifyMomentConvert[CentralMoment[2], FactorialMoment]variance = Factor[Limit[D[fmgf, {t, 2}], t -> 1] + mean(1 - mean)]属性和关系 (3)
FactorialMomentGeneratingFunction 等价于 
的 Expectation：
Expectation[t ^ n, nGeometricDistribution[p]]FactorialMomentGeneratingFunction[GeometricDistribution[p], t]Simplify[% - %%]FactorialMomentGeneratingFunction[GeometricDistribution[p], z]Sum[PDF[GeometricDistribution[p], k]z^k, {k, 0, Infinity}]Simplify[% == %%]𝒟 = HypergeometricDistribution[n, m, nt];D[FactorialMomentGeneratingFunction[𝒟, t], {t, 3}] /. t -> 1FactorialMoment[𝒟, 3]//Refine[#, nt ≥ 3]&%% - %//Simplify或者，也可使用 SeriesCoefficient：
SeriesCoefficient[3!FactorialMomentGeneratingFunction[𝒟, t], {t, 1, 3}]FactorialMoment[𝒟, 3]//Refine[#, nt ≥ 3]&%% - %//Simplify可能存在的问题 (2)
FactorialMoment[BetaNegativeBinomialDistribution[7, 11, 13], r]FactorialMomentGeneratingFunction[BetaNegativeBinomialDistribution[7, 11, 13], t]FactorialMomentGeneratingFunction 的解析式表示不总是已知的：
FactorialMomentGeneratingFunction[WalleniusHypergeometricDistribution[n, m, nt, w], t]巧妙范例 (1)
dists = {NegativeBinomialDistribution[10, 2 / 3], PoissonDistribution[3], BorelTannerDistribution[5 / 6, 10], ExponentialDistribution[1], BirnbaumSaundersDistribution[1, 3], HyperbolicDistribution[2, 1, 1, 2]};Table[Plot3D[Re[FactorialMomentGeneratingFunction[𝒟, x + I y]]//Evaluate, {x, -2, 2}, {y, -4, 4}, Mesh -> None, ImageSize -> 200, PlotLabel -> 𝒟], {𝒟, dists}]
Wolfram Research (2010)，FactorialMomentGeneratingFunction，Wolfram 语言函数，https://reference.wolfram.com/language/ref/FactorialMomentGeneratingFunction.html.
文本
Wolfram Research (2010)，FactorialMomentGeneratingFunction，Wolfram 语言函数，https://reference.wolfram.com/language/ref/FactorialMomentGeneratingFunction.html.
CMS
Wolfram 语言. 2010. "FactorialMomentGeneratingFunction." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/FactorialMomentGeneratingFunction.html.
APA
Wolfram 语言. (2010). FactorialMomentGeneratingFunction. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/FactorialMomentGeneratingFunction.html 年