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# 数学代写|ST302 Stochastic processes

## ST302课程简介

Availability

This course is compulsory on the BSc in Actuarial Science. This course is available on the BSc in Business Mathematics and Statistics, BSc in Data Science, BSc in Financial Mathematics and Statistics, BSc in Mathematics with Economics and BSc in Mathematics, Statistics and Business. This course is available as an outside option to students on other programmes where regulations permit and to General Course students.

Pre-requisites

Students must have completed either Probability, Distribution Theory and Inference (ST202) or Probability and Distribution Theory (ST206).

## Prerequisites

Teaching

This course will be delivered through a combination of classes, lectures, and Q&A sessions totalling a minimum of 29 hours across Michaelmas Term.

The course includes a reading week in Week 6 of Michaelmas Term.

Formative coursework

Compulsory written answers to two sets of problems.

Lecture notes will be provided. Relevant books include R Durrett, Essentials of Stochastic Processes; T Mikosch, Elementary Stochastic Calculus with Finance in View; Institute of Actuaries core reading notes.

## ST302 Stochastic processes HELP（EXAM HELP， ONLINE TUTOR）

If $X$ and $Y$ are independent binomial random variables with identical parameters $n$ and $p$, calculate the conditional expected value of $X$ given that $X+Y=m$.

A miner is trapped in a mine containing 3 doors. The first door leads to a tunnel that will take him to safety after 3 hours of travel. The second door leads to a tunnel that will return him to the mine after 5 hours of travel. The third door leads to a tunnel that will return him to the mine after 7 hours. If we assume that the miner is at all times equally likely to choose any one of the doors, what is the expected length of time until he reaches safety?

Consider $\mathrm{n}$ independent trials, each of which results in one of the outcomes ${1, \ldots, k}$, with respective probabilities $\left{p_1, \ldots, p_k\right}, \sum_{i=1}^k p_i=1$. Let $N_i$ denote the number of trials that result in outcome $i$, $i=1, \ldots, k$. For $i \neq j$ find $\mathbb{E}\left[N_i \mid N_j>0\right]$.

Let $U$ be a uniform random variable on $(0,1)$, and suppose that the conditional distribution of $X$, given that $U=p$, is binomial with parameters $n$ and $p$. Find the probability mass function of $X$. That is find for all $0 \leq i \leq n, \mathbb{P}(X=i)=$ ?

Hint: In the solution of this problem you may want to use the following general formula: Let $E$ be an event and $Y$ be a continuous r.v. with density function: $f_Y(y)$. Then:
$$\mathbb{P}(E)=\int_{-\infty}^{\infty} P(E \mid Y=y) f_Y(y) d y .$$
Moreover you may also want to use the following formula:
$$\int_0^1 p^i(1-p)^{n-i} d p=\frac{i !(n-i) !}{(n+1) !}$$

The joint density of $\mathrm{X}$ and $\mathrm{Y}$ is given by $f(x, y)=\frac{e^{-x / y} e^{-y}}{y}, \quad 0<x<\infty, \quad 0<y<\infty$. Compute $\mathbb{E}\left[X^2 \mid Y\right]=?$

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