CBMS-2015-11

题目来源：Question 11 日期：2024-08-01 题目主题：Math-概率论-概率生成函数

解题思路

这道题目主要涉及概率生成函数的性质和应用。我们需要:

1. 证明概率生成函数的基本性质
2. 推导独立同分布随机变量和的概率生成函数
3. 使用条件期望和全期望公式推导复合随机变量的概率生成函数
4. 应用概率生成函数的性质计算具体分布的期望

关键是要理解概率生成函数的定义和性质，以及如何利用这些性质来解决复杂的概率问题。

Solution

1. Proving basic properties of probability generating functions

(a) To show ${\phi }_X(1)=1$:

$$\begin{array}{rl}{\phi }_X(1)& =E\{1^X\}=\sum _{k=0}^{\infty }{1}^k\mathrm{Pr}\{X=k\}\\ & =\sum _{k=0}^{\infty }\mathrm{Pr}\{X=k\}\\ & =1\end{array}$$

The last step follows from the fact that the sum of probabilities over all possible outcomes is 1.

(b) To show $\frac{d{\phi }_X}{ds}(1)=E\{X\}$:

$$\begin{array}{rl}\frac{d{\phi }_X}{ds}(s)& =\frac{d}{ds}E\{s^X\}=\frac{d}{ds}\sum _{k=0}^{\infty }{s}^k\mathrm{Pr}\{X=k\}\\ & =\sum _{k=0}^{\infty }k{s}^{k-1}\mathrm{Pr}\{X=k\}\end{array}$$

Evaluating at $s=1$:

$$\begin{array}{rl}\frac{d{\phi }_X}{ds}(1)& =\sum _{k=0}^{\infty }k\cdot 1^{k-1}\mathrm{Pr}\{X=k\}\\ & =\sum _{k=0}^{\infty }k\mathrm{Pr}\{X=k\}\\ & =E\{X\}\end{array}$$

2. Probability generating function of sum of i.i.d. random variables

We need to show ${\phi }_{Y_n}(s)={\phi }_N(s{)}^n$ where $Y_n={\sum }_{i=1}^n{U}_i$.

$$\begin{array}{rl}{\phi }_{Y_n}(s)& =E\{s^{Y_n}\}=E\{s^{{\sum }_{i=1}^n{U}_i}\}\\ & =E\{s^{U_1}\cdot s^{U_2}\cdot ...\cdot s^{U_n}\}\\ & =E\{s^{U_1}\}\cdot E\{s^{U_2}\}\cdot ...\cdot E\{s^{U_n}\}\text{(due to i.i.d)}\\ & =(E\{s^N\}{)}^n\text{(due to i.i.d)}\\ & ={\phi }_N(s{)}^n\end{array}$$

3. Probability generating function of compound distribution

We need to show ${\phi }_W(s)={\phi }_N({\phi }_U(s))$ where $W={\sum }_{n=1}^{\infty }{Y}_{n}I(N=n)$.

Using the hint and the definition of probability generating function:

$$\begin{array}{rl}{\phi }_W(s)& =E\{s^W\}=\sum _{k=0}^{\infty }{s}^k\mathrm{Pr}\{W=k\}\\ & =\sum _{k=0}^{\infty }{s}^k\sum _{n=1}^{\infty }\mathrm{Pr}\{Y_n=k\}\mathrm{Pr}\{N=n\}\\ & =\sum _{n=1}^{\infty }\mathrm{Pr}\{N=n\}\sum _{k=0}^{\infty }{s}^k\mathrm{Pr}\{Y_n=k\}\\ & =\sum _{n=1}^{\infty }\mathrm{Pr}\{N=n\}{\phi }_{Y_n}(s)\\ & =\sum _{n=1}^{\infty }\mathrm{Pr}\{N=n\}{\phi }_N(s{)}^n\text{(from result of part 2)}\\ & ={\phi }_N({\phi }_N(s))\end{array}$$

4. Calculating expectations for specific distribution

Given $\mathrm{Pr}\{N=k\}=q^k(1-q)$, $(0<q<1)$, we need to calculate $E\{N\}$ and $E\{W\}$.

First, let’s calculate $E\{N\}$:

$$\begin{array}{rl}E\{N\}& =\sum _{k=0}^{\infty }k\mathrm{Pr}\{N=k\}=\sum _{k=0}^{\infty }k{q}^k(1-q)\\ & =(1-q)\sum _{k=0}^{\infty }k{q}^k=(1-q)\frac{q}{(1-q{)}^2}=\frac{q}{1-q}\end{array}$$

Now, for $E\{W\}$, we can use the result from part 1(b) and part 3:

$$\begin{array}{rl}E\{W\}& =\frac{d{\phi }_W}{ds}(1)=\frac{d}{ds}{\phi }_N({\phi }_N(s)){|}_{s=1}\\ & ={\phi }_N^{\prime }({\phi }_N(1))\cdot {\phi }_N^{\prime }(1)\\ & =E\{N{\}}^2\text{(using part 1(b))}\\ & =(\frac{q}{1-q}{)}^2\end{array}$$

Therefore, $E\{W\}=(\frac{q}{1-q}{)}^2$.

知识点

概率论概率生成函数条件期望全期望公式复合分布

难点思路

1. 理解概率生成函数的定义和基本性质
2. 利用独立性推导和的概率生成函数
3. 使用条件期望和全期望公式推导复合随机变型的概率生成函数
4. 应用概率生成函数的性质计算具体分布的期望

解题技巧和信息

1. 概率生成函数的基本性质：
   • ${\phi }_X(1)=1$
   • $\frac{d{\phi }_X}{ds}(1)=E\{X\}$
   • ${\phi }_{X+Y}(s)={\phi }_X(s)\cdot {\phi }_Y(s)$ (对于独立的 $X$ 和 $Y$)
2. 几何分布的概率生成函数：如果 $X\sim Geo(p)$，则 ${\phi }_X(s)=\frac{p}{1-(1-p)s}$
3. 利用全期望公式：$E\{Y\}=E\{E\{Y|X\}\}$

重点词汇

• Probability generating function: 概率生成函数
• Independent and identically distributed (i.i.d.): 独立同分布
• Compound distribution: 复合分布
• Conditional expectation: 条件期望
• Law of total expectation: 全期望公式
• Geometric distribution: 几何分布