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東京大学 新領域創成科学研究科 メディカル情報生命専攻 2017年8月実施 問題11

Author​

zephyr

Description​

Assume that the distributions of real-valued mutually independent random variables are identical and denoted as .

Denote by the random variables obtained by arranging in ascending order. Answer the following questions.

(1) Find the distribution function of .

(2) Find the distribution function of .

(3) Find the distribution function of for any .

(4) Find the expectation of when is the uniform distribution over .

假设实值相互独立随机变量 的分布是相同的，并记为 。

记 为将 按升序排列后得到的随机变量。回答以下问题。

(1) 找到 的分布函数。

(2) 找到 的分布函数。

(3) 找到任意 的 的分布函数。

(4) 当 为 上的均匀分布时，找到 的期望。

Kai​

(1)​

To find the distribution function of the smallest order statistic , we consider:

Since is the smallest of the , means that all . Thus:

We know that if and only if all , so:

Thus, the distribution function of is:

(2)​

To find the distribution function of the largest order statistic , we consider:

Since is the largest of the , means that at least one . Thus:

(3)​

To find the distribution function of the -th order statistic , we need to determine the probability . This represents the probability that the -th smallest value among is less than or equal to .

Step 1: Basic Concepts and Binomial Probability​

Since are independent and identically distributed, the probability that any particular is less than or equal to is . Similarly, the probability that is greater than is .

Step 2: Using Binomial Distribution​

We can think of this as a binomial distribution problem. We need to consider the event that at least out of values are less than or equal to . Mathematically, this can be expressed as:

Here, is the binomial coefficient, representing the number of ways to choose successes (values ) out of trials.

(4)​

If is the uniform distribution over , then for . Therefore:

The expectation of is given by:

where is the derivative of :

Therefore:

This is a Beta distribution integral:

Using the Beta function property, we get:

Thus:

Knowledge​

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 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 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.