16.3 Problems on Conditional Independence, Given a Random Vector

The pair $\left\{X , Y\right\} ci \left|\right. H$. $X \sim$ exponential $\left(u / 3\right)$, given $H = u$; $Y \sim$ exponential $\left(u / 5\right)$, given $H = u$; and $H \sim$ uniform $\left[1 , 2\right]$. Determine a general formula for $P \left(X > r , Y > s\right)$, 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 two additional members “at large.” Prior information indicates that this proportion is about 0.6 = 3/5. From a Bayesian point of view, the population proportion is taken to be the value of a random variable H. It seems reasonable to assume a…

Let $\left\{X_{i} : 1 \leq i \leq n\right\}$ be a random sample, given H. Set $W = \left(X_{1} , X_{2} , \hdots , X_{n}\right)$. Suppose X conditionally geometric $\left(u\right)$, given $H = u;$ i.e., suppose $P \left(\right. X = k \left|\right. H = u \left.\right) = u \left(1 - u\right)^{k}$ for all $k \geq 0$. If $H \sim$ uniform
on [0, 1], determine the best mean square estimator for H, given W.

Let $\left\{X_{i} : 1 \leq i \leq n\right\}$ be a random sample, given H. Set $W = \left(X_{1} , X_{2} , \hdots , X_{n}\right)$. Suppose X conditionally Poisson $\left(u\right)$, given $H = u;$ i.e., suppose $P \left(\right. X = k \left|\right. H = u \left.\right) = e^{- u} u^{k} / k !$. If $H \sim$ gamma $\left(m , \lambda\right)$, determine the best mean square estimator for H, given W.

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