16.2 Problems on conditional independence, given a random vector

The pair $\{X,Y\}\mathrm{ci}|H$ . $X\sim$ exponential $(u/3)$ , given $H=u$ ; $Y\sim$ exponential $(u/5)$ , given $H=u$ ; and $H\sim$ uniform $[1,2]$ . Determine a general formula for $P(X>r,Y>s)$ , then evaluate for $r=3,s=10$ .

A small random sample of size $n=12$ is taken to determine the proportion of the student body which favors a proposal to expand the student Honor Council by adding twoadditional members “at large.” Prior information indicates that this proportion is about 0.6 = 3/5. From a Bayesian point of view, the populationproportion is taken to be the value of a random variable H . It seems reasonable to assume a prior distribution $H\sim$ beta $(4,3)$ , giving a maximum of the density at $(4-1)/(4+3-2)=3/5$ . Seven of the twelve interviewed favor the proposition. What is the best mean-square estimate of the proportion, given this result? What is the conditionaldistribution of H , given this result?

Let $\{X_i:1\le i\le n\}$ be a random sample, given H . Set $W=(X_1,X_2,\cdots ,X_n)$ . Suppose X conditionally geometric $\left(u\right)$ , given $H=\mathrm{u;}$ i.e., suppose $P(X=k|H=u)=u{(1-u)}^k$ for all $k\ge 0$ . If $H\sim$ uniform
on [0, 1], determine the best mean square estimator for H , given W .

Let $\{X_i:1\le i\le n\}$ be a random sample, given H . Set $W=(X_1,X_2,\cdots ,X_n)$ . Suppose X conditionally Poisson $\left(u\right)$ , given $H=\mathrm{u;}$ i.e., suppose $P(X=k|H=u)=e^{-u}{u}^k/k!$ . If $H\sim$ gamma $(m,\lambda )$ , determine the best mean square estimator for H , given W .

Suppose $\{N,H\}$ is independent and $\{N,Y\}\mathrm{ci}|H$ . Use properties of conditional expectation and conditional independence to show that

Consider the composite demand D introduced in the section on Random Sums in "Random Selecton"

Suppose $\{N,H\}$ is independent, $\{N,Y_i\}\mathrm{ci}|H$ for all i , and $E\left[Y_i\right|H]=e\left(H\right)$ , invariant with i . Show that $E\left[D\right|H]=E[N\left]E\right[Y\left|H\right]\mathrm{a}.\mathrm{s}.$ .

The transition matrix P for a homogeneous Markov chain is as follows (in m-file npr16_07.m ):

1. Obtain the absolute values of the eigenvalues, then consider increasing powers of P to observe the convergence to the long run distribution.
2. Take an arbitrary initial distribution $p0$ (as a row matrix). The product $p0*P^k$ is the distribution for stage k . Note what happens as k becomes large enough to give convergence to the long run transition matrix. Does the endresult change with change of initial distribution $p0$ ?