数学代写|ST302 Stochastic processes

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 $\mathbb{E}\left[X^2 \mid Y\right]=?$