WOLFRAM

Wolfram 语言与系统参考资料中心

AsymptoticExpectation[expr,xdist,aa0]

假设 x 遵循概率分布 dist，计算中心在 a0 的表达式 expr 期望的渐近近似.

AsymptoticExpectation[expr,{x1,x2,…}dist,aa0]

假设 {x1,x2,…} 遵循多元分布 dist，计算中心在 a0 的表达式 expr 期望的渐近近似.

AsymptoticExpectation[expr,vars,{a,a0,n}]

计算 n 阶渐近期望.

更多信息和选项
Details and Options Details and Options
范例  
基本范例  
范围  
应用  
属性和关系  
参见
相关指南
历史
按以下格式引用：

AsymptoticExpectation

AsymptoticExpectation[expr,xdist,aa0]

假设 x 遵循概率分布 dist，计算中心在 a0 的表达式 expr 期望的渐近近似.

AsymptoticExpectation[expr,{x1,x2,…}dist,aa0]

假设 {x1,x2,…} 遵循多元分布 dist，计算中心在 a0 的表达式 expr 期望的渐近近似.

AsymptoticExpectation[expr,vars,{a,a0,n}]

计算 n 阶渐近期望.

更多信息和选项

范例

打开所有单元 关闭所有单元

基本范例  (2)

计算一个表达式期望的首个渐近近似：

Wolfram Language code: AsymptoticExpectation[2x ^ 2 + 3, xNormalDistribution[5a, E ^ a], a -> 0]
Wolfram Language code: Plot[Table[PDF[NormalDistribution[5a, E ^ a], x], {a, 0.1, 0.5, 0.1}]//Evaluate, {x, -3, 7}, PlotRange -> All, Filling -> Axis]

获取更高阶的近似：

Wolfram Language code: AsymptoticExpectation[2x ^ 2 + 3, xNormalDistribution[5a, E ^ a], {a, 0, 2}]

计算一个分布中生成矩量函数的渐近近似：

Wolfram Language code: AsymptoticExpectation[E ^ (t x), xGammaDistribution[α, β], {t, 0, 3}]

获取分布的前三个矩量：

Wolfram Language code: Table[Coefficient[%, t, k] * k!, {k, 1, 3}]//Factor
Wolfram Language code: Table[Moment[GammaDistribution[α, β], k], {k, 1, 3}]

范围  (9)

计算一个单变量连续分布的近似期望：

Wolfram Language code: AsymptoticExpectation[(x - b) ^ 2 + 5UnitStep[x ^ 2 - 1], xNormalDistribution[0, b], {b, 1, 2}]

单变量离散分布：

Wolfram Language code: AsymptoticExpectation[E ^ (d x) + 3, xPoissonDistribution[d], {d, 0, 5}]

多变量连续分布：

Wolfram Language code: AsymptoticExpectation[x ^ b + 3y, {x, y}ProductDistribution[NormalDistribution[0, 1], ExponentialDistribution[b]], {b, 0, 2}]

多变量离散分布：

Wolfram Language code: AsymptoticExpectation[2x + 3y + E ^ (b z), {x, y, z}MultinomialDistribution[10, {1 / 2, 1 / 3, 1 / 6}], {b, 0, 3}]

为 TransformedDistribution 计算渐近期望：

Wolfram Language code: AsymptoticExpectation[x ^ 3, xTransformedDistribution[E ^ (-a y ^ 2), yExponentialDistribution[1]], {a, ∞, 4}]

与正态分布的混合：

Wolfram Language code: AsymptoticExpectation[E ^ (-x), xMixtureDistribution[{1, 2}, {NormalDistribution[2, a], NormalDistribution[4, a]}], {a, 0, 5}]

参数混合分布：

Wolfram Language code: AsymptoticExpectation[x, xParameterMixtureDistribution[ExponentialDistribution[λ], λBetaDistribution[α, α ^ 2]], α -> Infinity]

边缘分布：

Wolfram Language code: AsymptoticExpectation[x^2 + E ^ x, xMarginalDistribution[DirichletDistribution[{1, 2, c}], 1], c -> 0]

方程式分布：

Wolfram Language code: AsymptoticExpectation[E ^ (-b x ^ 2) + 2, xProbabilityDistribution[6 / 5(x ^ 2 + 2x), {x, 0, 1}], {b, ∞, 2}]

应用  (5)

计算期望的渐近近似：

Wolfram Language code: AsymptoticExpectation[E ^ (-α x ^ 2), xExponentialDistribution[3], {α, 0, 3}, Assumptions -> α > 0]

与数字近似的结果比较：

Wolfram Language code: % /. {α -> 0.05}
Wolfram Language code: NExpectation[E ^ (-0.05 x ^ 2), xExponentialDistribution[3]]

计算一个分布平均值的渐近近似：

Wolfram Language code: AsymptoticExpectation[x, xHoytDistribution[q, ω], {q, 0, 2}, Assumptions -> q > 0]

精算一个分布方差的渐近近似：

Wolfram Language code: dist = RiceDistribution[α, β];
Wolfram Language code: AsymptoticExpectation[(x - Mean[dist]) ^ 2, xdist, {α, 0, 3}, Assumptions -> α > 0]

计算一个分布的矩量生成函数的渐近近似：

Wolfram Language code: dist = RayleighDistribution[σ];
Wolfram Language code: AsymptoticExpectation[E ^ (t x), xdist, {t, 0, 3}]

获取该分布的前三个矩量：

Wolfram Language code: Table[Coefficient[%, t, k] * k!, {k, 1, 3}]//Factor
Wolfram Language code: Table[Moment[dist, k], {k, 1, 3}]

计算一个二项分布的矩量生成函数的渐近近似：

Wolfram Language code: Block[{n = 20, p = 1 / 2}, Show[ DiscretePlot[PDF[BinomialDistribution[n, p], k]//Evaluate, {k, n}, PlotRange -> All, PlotMarkers -> Automatic], Plot[PDF[NormalDistribution[n p, Sqrt[n p - n p^2]], x], {x, -15, 25}, PlotStyle -> Thick]]]
Wolfram Language code: a1 = AsymptoticExpectation[E ^ (t x), xBinomialDistribution[n, p], {t, 0, 7}];

计算对应正态分布的矩量生成函数的近似：

Wolfram Language code: a2 = AsymptoticExpectation[E ^ (t x), xNormalDistribution[n p, Sqrt[n p - n p^2]], {t, 0, 7}];

对比 n 取较大值时的近似：

Wolfram Language code: a1 /. {n -> 60., p -> 0.3}
Wolfram Language code: a2 /. {n -> 60., p -> 0.3}

属性和关系  (4)

计算连续分布中一个表达式的渐近期望：

Wolfram Language code: dist = UniformDistribution[{-1, 2}];
Wolfram Language code: AsymptoticExpectation[E ^ (a x), xdist, {a, 0, 3}]

使用 AsymptoticIntegrate 获取相同结果：

Wolfram Language code: AsymptoticIntegrate[E ^ (a x) PDF[dist, x], {x, -∞, ∞}, {a, 0, 3}]

计算离散分布中一个表达式的渐近期望：

Wolfram Language code: dist = DiscreteUniformDistribution[{-2, 11}];
Wolfram Language code: AsymptoticExpectation[E ^ (a x), xdist, {a, 0, 3}]

使用 AsymptoticSum 获取相同结果：

Wolfram Language code: AsymptoticSum[E ^ (a x) PDF[dist, x], {x, -∞, ∞}, {a, 0, 3}]

使用 NExpectation 求期望的数值：

Wolfram Language code: dist[b_] := ExponentialDistribution[b];
Wolfram Language code: AsymptoticExpectation[E ^ (-x ^ 2), xdist[b], {b, 0, 6}]
Wolfram Language code: % /. {b -> 0.14}
Wolfram Language code: NExpectation[E ^ (-x ^ 2), xdist[0.14]]

使用 Expectation 求期望的精确值：

Wolfram Language code: dist[b_] := ExponentialDistribution[b];
Wolfram Language code: AsymptoticExpectation[E ^ (-x ^ 3), xdist[b], {b, 0, 6}]
Wolfram Language code: Expectation[E ^ (-x ^ 3), xdist[b]]

使用 Asymptotic 获取渐近近似：

Wolfram Language code: Asymptotic[%, {b, 0, 6}]
Wolfram Research (2020)，AsymptoticExpectation，Wolfram 语言函数，https://reference.wolfram.com/language/ref/AsymptoticExpectation.html.

文本

Wolfram Research (2020)，AsymptoticExpectation，Wolfram 语言函数，https://reference.wolfram.com/language/ref/AsymptoticExpectation.html.

CMS

Wolfram 语言. 2020. "AsymptoticExpectation." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/AsymptoticExpectation.html.

APA

Wolfram 语言. (2020). AsymptoticExpectation. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/AsymptoticExpectation.html 年

BibTeX

@misc{reference.wolfram_2026_asymptoticexpectation, author="Wolfram Research", title="{AsymptoticExpectation}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/AsymptoticExpectation.html}", note=[Accessed: 13-August-2026]}

BibLaTeX

@online{reference.wolfram_2026_asymptoticexpectation, organization={Wolfram Research}, title={AsymptoticExpectation}, year={2020}, url={https://reference.wolfram.com/language/ref/AsymptoticExpectation.html}, note=[Accessed: 13-August-2026]}