矩生成函数

Math 概率论

创建时间:2019-09-01 15:01

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定义
性质
高斯分布相关
参考

定义

参考维基百科 Moment Generating Functions

性质

LetX bearandomvariableswhosem.g.f.ψ(t)isfiniteforallvaluesoft insomeopen
interval around the point t = 0. Then, for each integer n > 0, the nth moment of X,
E(Xn), is finite and equals the nth derivative ψ(n)(t) at t = 0. That is, E(Xn) = ψ(n)(0)
for n = 1, 2, . . . .
$\begin{aligned} \psi^{(n)}(0) &=\left[\frac{d^{n}}{d t^{n}} E\left(e^{t X}\right)\right]_{t=0}=E\left[\left(\frac{d^{n}}{d t^{n}} e^{t X}\right)_{t=0}\right] \\ &=E\left[\left(X^{n} e^{t X}\right)_{t=0}\right]=E\left(X^{n}\right) \end{aligned}$
或者通过维基百科上的式子求导更容易理解
$\begin{aligned} M_{X}(t) &=\int_{-\infty}^{\infty} e^{t x} f(x) \mathrm{d} x \\ &=\int_{-\infty}^{\infty}\left(1+t x+\frac{t^{2} x^{2}}{2 !}+\cdots\right) f(x) \mathrm{d} x \\ &=1+t m_{1}+\frac{t^{2} m_{2}}{2 !}+\cdots \end{aligned}$
Let X be a random variable for which the m.g.f. is ψ 1 ; letY = aX + b, where a and b
are given constants; and let ψ 2 denote the m.g.f. of Y. Then for every value of t such
that ψ 1 (at) is finite,
$\psi_{2}(t)=e^{b t} \psi_{1}(a t)$
Suppose that X1 , . . . , Xn are n independent random variables; and for i = 1, . . . , n,
letψi denote the m.g.f.of X .LetY =X1 + . . . +Xn,and let the m.g.f.of Y be denoted
by ψ. Then for every value of t such that ψi(t) is finite for i = 1, . . . , n,
$\psi(t)=\prod_{i=1}^{n} \psi_{i}(t)$
If the m.g.f.’s of two random variables X1 and X2 are finite and identical for all values
of t in an open interval around the point t = 0, then the probability distributions of
X1 and X2 must be identical.
两个随机变量t=0领域内矩生成函数一样，那么他们的概率分布也是一样的。

高斯分布相关

高斯分布的矩生成函数可以结合上面定理4证明高斯分布很多其他性质

$\psi(t)=\exp \left(\mu t+\frac{1}{2} \sigma^{2} t^{2}\right) \quad \text { for }-\infty<t<\infty$

参考

书籍： Probability and Statistics 第四版 by Morris H. DeGroot

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