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 2016年8月実施 問題11

Author

zephyr

Description

Let be a sequence of mutually independent random variables such that each takes the value of 1 with probability , and 0 with probability (). We define a sequence () as follows:

Here, is a positive real number. Answer the following questions.

(1) Find the expected value of .

(2) Find the variance of .

(3) Express as a function of and the elements of ().

(4) Find ().

(5) Find as a function of and .

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设 为一组相互独立的随机变量，使得每个 以概率 取值为 1， 以概率 取值为 0（）。我们定义一个序列 （）如下：

其中， 是一个正实数。回答以下问题。

(1) 求 的期望值 。

(2) 求 的方差 。

(3) 表示 作为 和 元素的函数（）。

(4) 求 （）。

(5) 求 ，作为 和 的函数。

Kai

(1)

Given:

Thus,

To find :

Since takes the value 1 with probability and 0 with probability , we have:

Therefore,

(2)

To find , we first compute :

Thus,

Since is a Bernoulli random variable:

The variance of is:

(3)

To find the general form of , we solve the recurrence relation:

Starting from , we have:

It can be observed that:

(4)

Using linearity of expectation:

Since for all :

The sum is a geometric series:

(5)

Let's consider the limit by first simplifying the expression for .

Given:

Let's combine the terms by putting them over a common denominator:

Simplifying the numerator:

Now, let's find the limit for different values of .

Case 1:

When , as , and grow exponentially, and the dominant term will be . Thus, the limit is:

Since grows much faster than and , we have:

Thus, the limit does not exist in a finite value; it diverges to infinity.

Case 2:

When , we have:

As , the expected value becomes:

Thus, the limit also does not exist in a finite value; it diverges to infinity.

Case 3:

When , as , both and approach 0. The terms involving and become negligible, and the limit can be simplified as:

This limit exists and is finite.

In summary:

• For , does not exist as a finite value (diverges to infinity).
• For , does not exist as a finite value (diverges to infinity).
• For , , which is finite.

Knowledge

随机过程 期望值 几何级数

重点词汇

• Expected value: 期望值
• Variance: 方差
• Geometric series: 几何级数

参考资料

1. Probability and Statistics for Engineering and the Sciences, Chap. 4