Math Problems Vault

Problem 3.7.1

Show the following:

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(a)

$\mathbb {E}\left[a Y+b Z \mid X\right] = a \mathbb {E}\left[Y \mid X\right]+b \mathbb {E}\left[Z \mid X\right]$ for $a, b \in \mathbb {R}$ ,

(b)

$\mathbb {E}\left[Y \mid X\right] \geq 0$ if $Y \geq 0$ ,

(c)

$\mathbb {E}\left[1 \mid X\right] = 1$ ,

(d)

if $X$ and $Y$ are independent then $\mathbb {E}\left[Y \mid X\right] = \mathbb {E}\left[Y\right]$ ,

(e)

('pull-through property') $\mathbb {E}\left[Y g(X) \mid X\right] = g(X) \mathbb {E}\left[Y \mid X\right]$ for any suitable function $g$ ,

(f)

('tower property') $\mathbb {E}\left[\mathbb {E}\left[Y \mid X, Z\right] \mid X\right] = \mathbb {E}\left[Y \mid X\right] = \mathbb {E}\left[\mathbb {E}\left[Y \mid X\right] \mid X, Z\right]$ .

Problem 3.7.2

Suppose that $X$ and $Y$ are discrete random variables, and that $\phi (X)$ and $\psi (X)$ are two functions of $X$ satisfying

\mathbb {E}\left[\phi (X) g(X)\right] = \mathbb {E}\left[\psi (X) g(X)\right] = \mathbb {E}\left[Y g(X)\right]

for any function $g$ for which all the expectations exist. Show that $\phi (X)$ and $\psi (X)$ are almost surely equal, in that $\mathbb {P}\left(\phi (X) = \psi (X)\right) = 1$ .

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Problem 3.7.3

Suppose that the conditional expectation of $Y$ given $X$ is defined as the (almost surely) unique function $\psi (X)$ such that $\mathbb {E}\left[\psi (X) g(X)\right] = \mathbb {E}\left[Y g(X)\right]$ for all functions $g$ for which the expectations exist. Show (a)-(f) of Exercise (3.7.1) above (with the occasional addition of the expression 'with probability $1$ ').

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Problem 3.7.4

How should we define $\operatorname {Var}\left[Y \mid X\right]$ , the conditional variance of $Y$ given $X$ ? Show that $\operatorname {Var}\left[Y\right] = \mathbb {E}\left[\operatorname {Var}\left[Y \mid X\right]\right]+\operatorname {Var}\left[\mathbb {E}\left[Y \mid X\right]\right]$ .

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Problem 3.7.5

(a)(b)

The lifetime of a machine (in days) is a random variable $T$ with mass function $f$ . Given that the machine is working after $t$ days, what is the mean subsequent lifetime of the machine when:

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(a)

$f(x) = (N+1)^{-1}$ for $x \in \left\{ 0,1, \ldots , N\right\}$ ,

(b)

$f(x) = 2^{-x}$ for $x = 1,2, \ldots$ (The first part of Problem (3.11.13) may be useful.)

Problem 3.7.6

Let $X_{1}, X_{2}, \ldots$ be identically distributed random variables with mean $\mu$ and variance $\sigma^{2}$ , and let $N$ be a random variable taking values in the non-negative integers and independent of the $X_{i}$ . Let $S = X_{1}+X_{2}+\cdots +X_{N}$ . Show that $\mathbb {E}\left[S \mid N\right] = \mu N$ , and deduce that $\mathbb {E}\left[S\right] = \mu \mathbb {E}\left[N\right]$ . Find $\operatorname {Var}\left[S\right]$ in terms of the first two moments of $N$ , using the conditional variance formula of Exercise (3.7.4).

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Problem 3.7.7

A factory has produced $n$ robots, each of which is faulty with probability $\phi$ . To each robot a test is applied which detects the fault (if present) with probability $\delta$ . Let $X$ be the number of faulty robots, and $Y$ the number detected as faulty. Assuming the usual independence, show that

\mathbb {E}\left[X \mid Y\right] = \left\{ n \phi (1-\delta )+(1-\phi ) Y\right\} /(1-\phi \delta )

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Problem 3.7.8

(a)(b)

Each child is equally likely to be male or female, independently of all other children.

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(a)

Show that, in a family of predetermined size, the expected number of boys equals the expected number of girls. Was the assumption of independence necessary?

(b)

A randomly selected child is male; does the expected number of his brothers equal the expected number of his sisters? What happens if you do not require independence?

Problem 3.7.9

Let $X$ and $Y$ be independent with mean $\mu$ . Explain the error in the following equation:

{ }^{\prime } \mathbb {E}\left[X \mid X+Y = z\right] = \mathbb {E}\left[X \mid X = z-Y\right] = \mathbb {E}\left[z-Y\right] = z-\mu ^{\prime }

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Problem 3.7.10

A coin shows heads with probability $p$ . Let $X_{n}$ be the number of flips required to obtain a run of $n$ consecutive heads. Show that $\mathbb {E}\left[X_{n}\right] = \sum_{k = 1}^{n} p^{-k}$ .

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Problem 3.7.11

Give a definition of the conditional covariance $\operatorname {Cov}\left[X, Y \mid Z\right]$ . Show that

\operatorname {Cov}\left[X, Y\right] = \mathbb {E}\left[\operatorname {cov}(X, Y \mid Z)\right]+\operatorname {Cov}\left[\mathbb {E}\left[X \mid Z\right], \mathbb {E}\left[Y \mid Z\right]\right]

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Problem 3.7.12

An urn contains initially $b$ blue balls and $r$ red balls, where $b, r \geq 2$ . Balls are drawn one by one without replacement. Show that the mean number of draws until the first colour drawn is first repeated equals 3.

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Problem 3.7.13

(a)(b)

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(a)

Let $X$ be uniformly distributed on $\left\{ 0,1, \ldots , n\right\}$ . Show that $\operatorname {Var}\left[X\right] = \frac{1}{12} n(n+2)$ .

(b)

A student sits two examinations, gaining $X$ and $Y$ marks, respectively. In the interests of economy and fairness, the examiner determines that $X$ shall be uniformly distributed on $\left\{ 0,1, \ldots , n\right\}$ , and that, conditional on $X = k, Y$ shall have the binomial $\operatorname {bin}(n, k / n)$ distribution.

(i) Show that $\mathbb {E}\left[Y\right] = \frac{1}{2} n$ , and $\operatorname {Var}\left[Y\right] = \frac{1}{12}\left(n^{2}+4 n-2\right)$ .

(ii) Find $\mathbb {E}\left[X+Y\right]$ and $\operatorname {Var}\left[X+Y\right]$ .

Problem 3.7.14

Let $X, Y$ be discrete integer-valued random variables with the joint mass function

f(x, y) = \frac{\lambda ^{y} e^{-2 \lambda }}{x!(y-x)!}, \quad 0 \leq x \leq y < \infty

Show that $X$ and $Y$ are each Poisson-distributed, and that the conditional distribution of $X$ given $Y$ is binomial.

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