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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 2022年8月実施 問題11

Author​

zephyr

Description​

Let be a random sequence of non-negative integers generated by the following rules.

(i) If , with probability , and with probability .

(ii) If , with probability 1.

In the following, is assumed. Further, we define as the probability that at time with initial value (: a nonnegative integer). Answer the following questions.

(1) Answer the probability that given .

(2) Answer the probability that given .

(3) Express using and (, ).

(4) Let . Derive the equations that the s satisfy using (3).

(5) Answer the condition for that the equations of (4) have a solution with , as well as the solution (Examine the case: ).

Kai​

(1) The probability that given ​

To find the probability that given , we need to consider the different paths the process can take to reach from 1 to 2 in three steps.

The paths and their probabilities are:

  1. : probability
  2. : probability

Adding these probabilities together, we get:

(2) The probability that given ​

To find the probability that given , we consider the paths to reach 0 from 2 in four steps.

The paths and their probabilities are:

  1. : probability
  2. : probability
  3. : probability

Adding these probabilities together, we get:

(3) Express using and (, )​

For , the probability that given can be written in terms of the probabilities at :

(4) Let . Derive the equations that the s satisfy using (3)​

As , becomes time-independent. Thus,

(5) The condition for that the equations of (4) have a solution with , as well as the solution (Examine the case: )​

Assume a solution of the form :

Dividing by , we get:

This is a quadratic equation in :

The roots of this equation are:

For the solution to converge to 0 as , we need the root with the negative sign:

Therefore, the solution is:

Knowledge​

随机过程 马尔可夫链 概率计算

难点解题思路​

  • 分析每个时间步的状态变化及其概率。
  • 考虑随机过程的限制条件如 时的吸收状态。

解题技巧和信息​

  • 分步计算状态转移概率。
  • 利用马尔可夫链的平稳状态来解答长时间行为问题。

重点词汇​

  • random sequence 随机序列
  • probability 概率
  • Markov chain 马尔可夫链
  • absorbing state 吸收状态

参考资料​

  1. Ross, S. M. (2007). Introduction to Probability Models. Chapter 4: Markov Chains.