Abstract: 本文介绍另一个基于期望的随机变量分布的数字特征——“矩” （不知道咋翻译的，英文moment表示时刻,瞬间等意思）
Keywords: Moments，Moments Generating Function

# 距

## 距的存在 Existence of Moments

• 如果随机变量有界，那么距存在（也就是如果有有限的数字 $a,b$ 使得 $Pr(a\leq X\leq b)=1$ 成立）
• 即使随机变量没有界，那么也有可能存在所有的距。
• 下面定理表示如果存在 $k$ 阶距，那么存在所有小于 $k$ 阶的距。

Theorem If $E(|X|^k)<\infty$ for some positive integer $k$ ,then $E(|X|^j)<\infty$ for every positive integer $j$ such that $j < k$

\begin {aligned} E(|X|^j)&=\int^{\infty}_{-\infty}|x|^jf(x)dx\\ &=\int_{|x|\leq 1}|x|^jf(x)dx+\int_{|x|>1}|x|^jf(x)dx\\ &\leq \int_{|x|\leq 1}1\cdot f(x)dx+\int_{|x|>1}|x|^kf(x)dx\\ &\leq Pr(|X|\leq 1)+E(|X|^k) \end{aligned}

### 中心距 Central Moments

Suppose that $X$ is a random variable for which $E(X)=\mu$ .For every positive integer $k$ ,the expectation $E [(X-\mu)^k]$ is called the $k$th central moment of $X$ or the $k$th moment of $X$ about the mean.

$$f(x)=ce^{-(x-3)^2/2} \text{ for } -\infty<x<\infty$$

### 偏度 Skewness

Definition Let $X$ be a random variable with mean $\mu$ ,standard deviation $\sigma$ ,and finite third moment. The skewness of $X$ is defined to be $E[(X-\mu)^3]/\sigma^3$

## 距生成函数 Moment Generating Functions

Definition Moment Generating Function. Let X be a random variable.For each real number t,define
$$\psi(t)=E(e^{tX})$$
The function $\psi(t)$ is called the moment generating function(abbreviated m.g.f.)of $X$

• 随机变量有界，所以 $e^{tX}$ 对所有 $t$ 都是有限的。
• 随机变量没有界，那么有些 $t$ 是有限的，有些 $t$ 则不是
• $t=0$ 这种特殊情况，m.g.f. 为1

Theorem Let $X$ be a random variables whose m.g.f $\psi(t)$ is finite for all values of $t$ in some open interval around the point $t=0$ .Then,for each integer $n>0$ ,the $n$th moment of $X$ ,$E(X)$,is finite and equals the $n$th derivative $\psi^{(n)}(t)$ at $t=0$ ,That is $E(X^n)=\psi^{(n)}(0)$ for $n=1,2\dots$

$$f(x)=\begin{cases} e^{-x} & \text{for } x>0\\ 0&\text{otherwise} \end{cases}$$

$$\psi(x)=E(e^{tX})=\int^{\infty}_{0}e^{tx}e^{-x}dx\\ =\int^{\infty}_{0}e^{(t-1)x}dx$$

$$\psi(t)=\frac{1}{1-t}$$

$$\psi’(t)=\frac{1}{(1-t)^2}\\ \psi’’(t)=\frac{2}{(1-t)^3}$$

$$Var(X)=\psi’’(0)-[\psi’(0)]^2=1$$

## 距生成函数的性质 Properties of Moment Generating Functions

Theorem Let $X$ be a random varibale for which the m.g.f. is $\psi_1$ ;Let $Y=aX+b$ ,where $a$ and $b$ are given constants; and let $\psi_2$ denote the m.g.f. of $Y$ .Then for every value of $t$ such that $\psi_1(at)$ is finite,
$$\psi_2(t)=e^{bt}\psi_1(at)$$

$$\psi_2(t)=E(e^{tY})=E[e^{(aX+b)t}]=e^{bt}E(e^{atX})=e^{bt}\psi_1(at)$$

Theorem Suppose that $X_1,\dots,X_n$ are $n$ independent random varibales;and for $i=1,\dots,n$ . let $\psi_i$ denote the m.g.f. of $X_i$ .Let $Y=X_1+\dots+X_n$ ,and let the m.g.f. of $Y$ be denoted by $\psi$ .Then for every value of t such that $\psi_i(t)$ is finite for $i=1,\dots,n$ ,
$$\psi(t)=\Pi^{n}_{i=1}\psi_i(t)$$

$$\psi(t)=E(e^{tY})=E(e^{tX_1+\dots+tX_n})=E(\Pi^{n}_{i=1}e^{tX_i})\\ E(\Pi^{n}_{i=1}e^{tX_i})=\Pi^{n}_{i=1}E(e^{tX_i})\\ \psi(t)=\Pi^{n}_{i=1}\psi_i(t)$$

### 二项分布的距生成函数 The Moment Generating Function for the Binomial Distribution

\begin{aligned} \psi_i(t)&=E(e^{tX_i})\\ &=p\times e^{t}+(1-p)\times e^{0}\\ &=pe^t+1-p \end{aligned}

$$\psi(t)=(pe^t+1-p)^n$$

### 距生成函数的唯一性 Uniqueness of Moment Generating Functions

Theorem If the m.g.f. of two random varibales $X_1$ and $X_2$ are finite and identical for all values of $t$ in an open interval around the point $t=0$ ,then the probability distributions of $X_1$ and $X_2$ must be identical.

### 二项分布的可加性 The Additive Property of the Binomial Distribution

Theorem If $X_1$ and $X_2$ are independent random variables,and if $X_i$ has the binomial distribution with parameters $n_i$ and $p$ ($i=1,2,\dots$ ),then $X_1+X_2$ has the binomial distirbution with parameters $n_1+n_2$ and $p$

$$\psi_i(t)=(pe^{t}+1-p)^{n_i}$$

$$\psi(t)=(pe^{t}+1-p)^{n_1+n_2}$$

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