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卡方分布的矩母函数是什么形式的

创建于：2025年8月15日

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问题

卡方分布的矩母函数是什么形式的

思考

Thought for 5s

回答

卡方分布 X∼χν2X\sim \chi^2_\nu（自由度 ν\nu）的矩母函数为

MX(t)=(1−2t)−ν/2,t<12.M_X(t)=(1-2t)^{-\nu/2},\qquad t<\tfrac12.

由 χν2≡Gamma(α=ν/2, θ=2)\chi^2_\nu \equiv \mathrm{Gamma}(\alpha=\nu/2,\ \theta=2) 且 Gamma 的 mgf 为 (1−θt)−α(1-\theta t)^{-\alpha} 得到上式。相应的累积母函数：

KX(t)=ln⁡MX(t)=−ν2ln⁡(1−2t).K_X(t) = \ln M_X(t)= -\frac{\nu}{2}\ln(1-2t).

顺带：由 mgf 可得 E[X]=ν\mathbb E[X]=\nu，Var(X)=2ν\mathrm{Var}(X)=2\nu。
（若是非中心卡方 X∼χν2(λ)X\sim \chi^2_\nu(\lambda)，则 MX(t)=exp⁡ ⁣(λt1−2t) (1−2t)−ν/2, t<12M_X(t)=\exp\!\big(\frac{\lambda t}{1-2t}\big)\,(1-2t)^{-\nu/2},\ t<\tfrac12。）

问题

t分布和f分布的矩母函数又是咋样的

思考

Exploring MGF of t- and F-Distributions

The MGF of the Student's t-distribution doesn't exist for any t ≠ 0 when df > 1 because the expected value of etXe^{tX} diverges. However, its characteristic function exists via a special function.

For the F-distribution with degrees of freedom d1 and d2, the MGF exists only when t<d2/2t < d2/2.

Deriving MGF for F-Distribution

For the F-distribution with d1 and d2 degrees of freedom, the MGF involves the Beta function. However, the integral for the MGF, E[etX]E[e^{tX}], likely diverges for t>0t > 0 due to heavy tails. It only seems to converge for negative tt, resembling a Laplace transform when t<0t < 0.

Clarifying MGF for t- and F-Distributions

For both the t- and F-distributions, the MGF does not exist for any positive t due to the heavy tail, which causes divergence. The MGF, as typically defined, requires existence in a neighborhood around zero, but it only seems to exist for negative t. In the case of the F-distribution, some sources even present it as a hypergeometric function but only for t<d22t < \frac{d2}{2}.

Clarifying MGF of Beta Prime vs F-Distribution

I think I was mixing up the F-distribution with the Beta Prime distribution's MGF. The F-distribution is closely related to the Beta prime, scaled by d2d1\frac{d2}{d1}. The MGF for Beta prime actually doesn’t exist for t>0t > 0 because the tail's growth, when combined with the exponential, causes divergence.

Explaining the absence of MGF for t and F distributions

For both the t-distribution and F-distribution, the moment generating function (MGF) doesn't exist because the integrals for t > 0 diverge due to the heavy-tailed distributions. The t-distribution has finite moments only for certain orders, and the F-distribution’s MGF does not converge in an open interval around 0.

Thought for 51s

回答

简要结论：

  • t 分布 X∼tνX\sim t_\nu：** 没有矩母函数（MGF）。除 t=0t=0 外，E[etX]\mathbb E[e^{tX}] 发散；只有当 ν>k\nu>k 时第 kk 阶矩才存在，因此不可能在 0 附近有可展开的 MGF。若需要，t 分布的特征函数**是存在的（可写成含改良贝塞尔 KK 函数的形式）。

  • F 分布 Y∼Fd1,d2Y\sim F_{d_1,d_2}：** 也没有矩母函数**（在 0 的任意邻域都不收敛）。其右尾仅作幂次衰减，导致对任意 t>0t>0 的 E[etY]\mathbb E[e^{tY}] 发散；并且仅当 d2>2kd_2>2k 时第 kk 阶矩才存在，这同样排除了 MGF 的存在。
    说明：对 非正 参数（把 t=−s≤0t=-s\le0 看作拉普拉斯变换）时，E[e−sY]\mathbb E[e^{-sY}] 是存在的，并可用超几何函数表达，但这不构成通常意义上的 MGF。

问题

那他们有特征函数吗

思考

Finalizing formulas

I’ll provide the formulas for the characteristic functions:

  • t-distribution CF: φ_X(t) = ( (√ν |t|)^{ν/2} K_{ν/2}(√ν |t|) ) / ( 2^{ν/2 - 1} Γ(ν/2) ). We might mention simpler forms with polynomials for integer ν, but that’s not necessary here.

  • F-distribution CF: φ_{F}(s) = Γ((d1 + d2)/2) / Γ(d2/2) * U(d1/2, 1 - d2/2, - (d2/d1) i s). I’ll note that the CF always exists and include citations accordingly.

Crafting response in Chinese

For the t-distribution, I’ll start with: "有。特征函数(φ)总是存在," and provide the formula with the modified Bessel K. I'll clarify that K_ν is the modified Bessel function of the second kind, and valid for all ν>0.

For the F-distribution, I’ll share the formula with the confluent hypergeometric U function (Tricomi). I'll add that the t-distribution is symmetric, with φ_X as a real, even function, and that its MGF diverges but the CF exists.

It’s important to mention that the F-distribution formula uses the Tricomi U, and many sources have incorrect formulas. I’ll include a reference to Phillips 1982. I’ll note no MGF exists, but the Laplace transform exists for s ≥ 0 in terms of the hypergeometric function. I'll keep it concise.

Writing concise response in Chinese

I'll begin with: "有，都有。" Then, I’ll proceed with formulas:

  • For the t-distribution: Write the formula φ_X(t) = ..., valid for all t ∈ ℝ. Mention K_ν as the modified Bessel function of the second kind. Cite Gaunt 2019.

  • For the F-distribution: Express φ_{d1,d2}^F(s) = Γ(...) / Γ(...) U(a,b,z) and expand z = - (d2/d1) i s. Include citations for Wikipedia and possibly JSTOR.

I’ll make sure it’s clear, concise, and without unnecessary language.

Thought for 1m 6s

回答

有的，二者的特征函数都存在（哪怕 MGF 不存在）：

  • t 分布 X∼tνX\sim t_\nu：

    φX(t)=E[eitX]=(ν ∣t∣)ν/2 Kν/2 ⁣(ν ∣t∣)2ν/2−1 Γ(ν/2) ,t∈R,\varphi_X(t)=\mathbb E[e^{itX}] =\frac{\big(\sqrt{\nu}\,|t|\big)^{\nu/2}\,K_{\nu/2}\!\big(\sqrt{\nu}\,|t|\big)}{2^{\nu/2-1}\,\Gamma(\nu/2)}\,,\quad t\in\mathbb R,

    其中 Kα(⋅)K_{\alpha}(\cdot) 为第二类改良贝塞尔函数。(arXiv)

  • F 分布 Y∼Fd1,d2Y\sim F_{d_1,d_2}：

    φY(t)=E[eitY]=Γ ⁣(d1+d22)Γ ⁣(d22) U ⁣(d12, 1−d22, − d2d1 it),t∈R,\varphi_Y(t)=\mathbb E[e^{itY}] =\frac{\Gamma\!\big(\tfrac{d_1+d_2}{2}\big)}{\Gamma\!\big(\tfrac{d_2}{2}\big)} \,U\!\left(\tfrac{d_1}{2},\,1-\tfrac{d_2}{2},\, -\,\tfrac{d_2}{d_1}\, i t\right),\quad t\in\mathbb R,

    其中 U(a,b,z)U(a,b,z) 为第二类合流超几何函数（Tricomi 函数）；不少教材里曾给出过错误形式，这一式为更正后的结果。(Wikipedia, JSTOR)

需要的话，我也可以给出它们在 t→0t\to 0 或 ∣t∣→∞|t|\to\infty 时的近似展开。

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