Riemann surface
matlab

还在探索随机过程的学习难题吗?别担心!我们的random-process-guide团队专业为您解决各种与随机过程相关的问题。我们拥有深厚的专业背景和丰富的经验,能够帮助您完成高水平的作业和论文,让您的学习之路一帆风顺!

以下是一些我们可以帮助您解决的问题:

基本随机过程:各种常用随机过程的定义、性质和分类,如马尔可夫链、泊松过程、随机游走等。

高级随机过程:更复杂的随机过程和算法,如布朗运动、随机矩阵、马尔可夫决策过程等。

随机过程建模和分析:常见的随机过程建模技巧和分析方法,如随机过程的稳定性、瞬时分析、平稳性等。

随机模拟和推断:使用模拟和推断方法对随机过程进行建模和预测,如蒙特卡洛方法、极大似然估计等。

随机过程优化:随机过程的优化问题,如最优控制、随机最优化、马尔可夫决策过程优化等。

随机过程与统计:随机过程与统计学的关系,如随机过程的相关性、随机过程的参数估计等。

无论您面临的随机过程问题是什么,我们都会尽力为您提供专业的帮助,确保您的学习之旅顺利无阻!

问题 1.


In the Dark Ages, Harvard, Dartmouth, and Yale admitted only male students. Assume that, at that time, 80 percent of the sons of Harvard men went to Harvard and the rest went to Yale, 40 percent of the sons of Yale men went to Yale, and the rest split evenly between Harvard and Dartmouth; and of the sons of Dartmouth men, 70 percent went to Dartmouth, 20 percent to Harvard, and 10 percent to Yale. (i) Find the probability that the grandson of a man from Harvard went to Harvard. (ii) Modify the above by assuming that the son of a Harvard man always went to Harvard. Again, find the probability that the grandson of a man from Harvard went to Harvard..


Solution. We first form a Markov chain with state space $S={H, D, Y}$ and the following transition probability matrix :
$$
\mathrm{P}=\left(\begin{array}{ccc}
.8 & 0 & .2 \
.2 & .7 & .1 \
.3 & .3 & .4
\end{array}\right)
$$
Note that the columns and rows are ordered: first $H$, then $D$, then $Y$. Recall: the $i j^{\text {th }}$ entry of the matrix $\mathrm{P}^n$ gives the probability that the Markov chain starting in state $i$ will be in state $j$ after $n$ steps. Thus, the probability that the grandson of a man from Harvard went to Harvard is the upper-left element of the matrix
$$
\mathrm{P}^2=\left(\begin{array}{ccc}
.7 & .06 & .24 \
.33 & .52 & .15 \
.42 & .33 & .25
\end{array}\right)
$$
It is equal to $.7=.8^2+.2 \times .3$ and, of course, one does not need to calculate all elements of $\mathrm{P}^2$ to answer this question.
If all sons of men from Harvard went to Harvard, this would give the following matrix for the new Markov chain with the same set of states:
$$
P=\left(\begin{array}{ccc}
1 & 0 & 0 \
.2 & .7 & .1 \
.3 & .3 & .4
\end{array}\right)
$$
The upper-left element of $\mathrm{P}^2$ is 1 , which is not surprising, because the offspring of Harvard men enter this very institution only.

问题 2.

A certain calculating machine uses only the digits 0 and 1 . It is supposed to transmit one of these digits through several stages. However, at every stage, there is a probability p that the digit that enters this stage will be changed when it leaves and a probability $q=1-p$ that it won’t. Form a Markov chain to represent the process of transmission by taking as states the digits 0 and 1 . What is the matrix of transition probabilities?
Now draw a tree and assign probabilities assuming that the process begins in state 0 and moves through two stages of transmission. What is the probability that the machine, after two stages, produces the digit 0 (i.e., the correct digit)?

Solution. Taking as states the digits 0 and 1 we identify the following Markov chain (by specifying states and transition probabilities):
$$
\begin{array}{lll}
0 & 1 \
1 & q & p \
1 & p & q
\end{array}
$$
where $p+q=1$. Thus, the transition matrix is as follows:
$$
\mathrm{P}=\left(\begin{array}{ll}
q & p \
p & q
\end{array}\right)=\left(\begin{array}{cc}
1-p & p \
p & 1-p
\end{array}\right)=\left(\begin{array}{cc}
q & 1-q \
1-q & q
\end{array}\right) .
$$
It is clear that the probability that that the machine will produce 0 if it starts with 0 is $p^2+q^2$.

E-mail: help-assignment@gmail.com  微信:shuxuejun

help-assignment™是一个服务全球中国留学生的专业代写公司
专注提供稳定可靠的北美、澳洲、英国代写服务
专注于数学,统计,金融,经济,计算机科学,物理的作业代写服务

发表回复

您的电子邮箱地址不会被公开。 必填项已用 * 标注