CBMS-2018-11

题目来源：[[做题/文字版题库/CBMS/2018#Question 11|2018#Question 11]] 日期：2024-07-27 题目主题：CS-概率论-顺序统计量

解题思路

我们需要找到顺序统计量的分布函数和期望值。主要知识点包括概率分布函数、累积分布函数、联合概率分布、以及期望值计算。解题过程中要利用独立随机变量的性质及其分布特性。

Solution

1. Distribution Function of ${\mathbf{X}}_{(1)}$

To find the distribution function of the smallest order statistic ${\mathbf{X}}_{(1)}$, we consider:

$$F_{{\mathbf{X}}_{(1)}}(x)=P({\mathbf{X}}_{(1)}\le x).$$

Since ${\mathbf{X}}_{(1)}$ is the smallest of the ${\mathbf{X}}_1,\dots ,{\mathbf{X}}_{\mathbf{n}}$, ${\mathbf{X}}_{(1)}>x$ means that all ${\mathbf{X}}_{\mathbf{i}}>x$. Thus:

$$F_{{\mathbf{X}}_{(1)}}(x)=1-P({\mathbf{X}}_{(1)}>x).$$

We know that ${\mathbf{X}}_{(1)}>x$ if and only if all ${\mathbf{X}}_{\mathbf{i}}>x$, so:

$$P({\mathbf{X}}_{(1)}>x)=P({\mathbf{X}}_1>x,{\mathbf{X}}_2>x,\dots ,{\mathbf{X}}_{\mathbf{n}}>x)={\left(P({\mathbf{X}}_1>x)\right)}^n=(1-F(x){)}^n.$$

Thus, the distribution function of ${\mathbf{X}}_{(1)}$ is:

$$F_{{\mathbf{X}}_{(1)}}(x)=1-(1-F(x){)}^n.$$

2. Distribution Function of ${\mathbf{X}}_{(\mathbf{n})}$

To find the distribution function of the largest order statistic ${\mathbf{X}}_{(\mathbf{n})}$, we consider:

$$F_{{\mathbf{X}}_{(\mathbf{n})}}(x)=P({\mathbf{X}}_{(\mathbf{n})}\le x).$$

Since ${\mathbf{X}}_{(\mathbf{n})}$ is the largest of the ${\mathbf{X}}_1,\dots ,{\mathbf{X}}_{\mathbf{n}}$, ${\mathbf{X}}_{(\mathbf{n})}\le x$ means that at least one ${\mathbf{X}}_{\mathbf{i}}\le x$. Thus:

$$F_{{\mathbf{X}}_{(\mathbf{n})}}(x)=P({\mathbf{X}}_1\le x,{\mathbf{X}}_2\le x,\dots ,{\mathbf{X}}_{\mathbf{n}}\le x)={\left(P({\mathbf{X}}_1\le x)\right)}^n=(F(x){)}^n.$$

3. Distribution Function of ${\mathbf{X}}_{(\mathbf{k})}$

To find the distribution function of the $k$-th order statistic ${\mathbf{X}}_{(\mathbf{k})}$, we need to determine the probability $F_{{\mathbf{X}}_{(\mathbf{k})}}(x)=P({\mathbf{X}}_{(\mathbf{k})}\le x)$. This represents the probability that the $k$-th smallest value among ${\mathbf{X}}_1,\dots ,{\mathbf{X}}_{\mathbf{n}}$ is less than or equal to $x$.

Step 1: Basic Concepts and Binomial Probability

Since ${\mathbf{X}}_{\mathbf{i}}$ are independent and identically distributed, the probability that any particular ${\mathbf{X}}_{\mathbf{i}}$ is less than or equal to $x$ is $F(x)$. Similarly, the probability that ${\mathbf{X}}_{\mathbf{i}}$ is greater than $x$ is $1-F(x)$.

Step 2: Using Binomial Distribution

We can think of this as a binomial distribution problem. We need to consider the event that at least $k$ out of $n$ ${\mathbf{X}}_{\mathbf{i}}$ values are less than or equal to $x$. Mathematically, this can be expressed as:

$$F_{{\mathbf{X}}_{(\mathbf{k})}}(x)=P({\mathbf{X}}_{(\mathbf{k})}\le x)=\sum _{j=k}^n\left(\genfrac{}{}{0ex}{}{n}{j}\right)(F(x){)}^j(1-F(x){)}^{n-j}.$$

Here, $\left(\genfrac{}{}{0ex}{}{n}{j}\right)$ is the binomial coefficient, representing the number of ways to choose $j$ successes (values $\le x$) out of $n$ trials.

4. Expectation of ${\mathbf{X}}_{(1)}$ when $F(x)$ is the Uniform Distribution over $[0,1]$

If $F(x)$ is the uniform distribution over $[0,1]$, then $F(x)=x$ for $x\in [0,1]$. Therefore:

$$F_{{\mathbf{X}}_{(1)}}(x)=1-(1-x{)}^n.$$

The expectation of ${\mathbf{X}}_{(1)}$ is given by:

$$\mathbb{E}[{\mathbf{X}}_{(1)}]={\int }_0^{1}x{f}_{{\mathbf{X}}_{(1)}}(x)\mathrm{d}x,$$

where $f_{{\mathbf{X}}_{(1)}}(x)$ is the derivative of $F_{{\mathbf{X}}_{(1)}}(x)$:

$$f_{{\mathbf{X}}_{(1)}}(x)=\frac{\mathrm{d}}{\mathrm{dx}}\left[1-(1-x{)}^n\right]=n(1-x{)}^{n-1}.$$

Therefore:

$$\mathbb{E}[{\mathbf{X}}_{(1)}]={\int }_0^{1}x\cdot n(1-x{)}^{n-1}\mathrm{d}x.$$

This is a Beta distribution integral:

$$\mathbb{E}[{\mathbf{X}}_{(1)}]=n{\int }_0^{1}x(1-x{)}^{n-1}\mathrm{d}x.$$

Using the Beta function property, we get:

$${\int }_0^{1}x(1-x{)}^{n-1}\mathrm{d}x=\frac{1}{n+1}.$$

Thus:

$$\mathbb{E}[{\mathbf{X}}_{(1)}]=\frac{n}{n+1}.$$

知识点

顺序统计量概率分布函数期望值 Beta分布

难点思路

第 3 小问关于任意 $k$ 阶顺序统计量的分布函数需要理解 Binomial 分布的性质并进行累加，这是一个较难点。

解题技巧和信息

对于顺序统计量，了解如何通过分布函数 $F(x)$ 来表示最小和最大顺序统计量的分布函数非常重要。对于均匀分布的情况，可以利用 Beta 分布性质简化期望值计算。

重点词汇

• order statistic 顺序统计量
• distribution function 分布函数
• expectation 期望值
• uniform distribution 均匀分布

参考资料

1. “Probability and Statistics” by Morris H. DeGroot and Mark J. Schervish, Chapter 5.
2. “A First Course in Probability” by Sheldon Ross, Chapter 8.