Conditional Expectation

A fair die with faces $1,2,3$ colored green and faces $4,5,6$ colored red is rolled once. Let $X$ be the number shown.

Show that $\mathbb {E}\left[ \; \mathbb {1}_{A} \mid B \right]=\mathbb {P}\left( A \mid B \right)$ if $\mathbb {P}\left(B\right) \neq 0$. (As usual, $\; \mathbb {1}_{A}$ denotes the indicator function of an event $A$.)

A fair die is rolled twice. What is the conditional expectation of the sum of outcomes given the first roll shows 1 ?

We toss a coin repeatedly until heads appear, hitting the numerical keypad of a calculator randomly after each toss. Find the conditional expectation of the number keyed in, given that heads appear for the first time after $i$ tosses. (Our calculator is a special model capable of displaying infinitely many digits.)

Let $X$ be a random variable with the Poisson distribution. Find the conditional expectation of $X$ given that $X$ is an even number.

For any discrete random variable $X$ and any event $B$ such that $\mathbb {P}\left(B\right) \neq 0$, show that

where $\; \mathbb {1}_{B}$ is the indicator function of $B$.

$X$ is the sum of outcomes obtained by rolling a die twice and $B_{i}$ is the event that the first die shows $i$.

$X$ denotes the amount shown if three coins, $5 \not\subset , 10$, and $25 \neq$, are tossed, and $B_{i}$ denotes the event that $i$ coins land heads up.

$X$ denotes the number of points scored by a chess player in a match against an equally strong opponent, where draws are discarded so that the odds of winning or losing a game are fifty-fifty, the match ends if one of the players reaches 3 points, and $B_{i}$ is the event that $i$ games have been played.

Let $X$ be a random variable with exponential distribution. Find $\mathbb {E}\left[ X \mid \{ X \geqslant t\} \right]$.

Let $X$ and $Y$ be jointly continuous random variables with joint density

Compute the conditional expectation of $X+Y$ given that $X<Y$.

Show that if $\mathbb {E}\left[X\right]$ is exists, then so does $\mathbb {E}\left[ X \mid B \right]$ for any event $B$ such that $\mathbb {P}\left(B\right) \neq 0$.

A die is rolled twice; $X$ is the sum of the outcomes and $Y$ is the outcome of the first roll.

Three coins, $5 \notin , 10 \not\phi$, and $25 \neq$, are tossed; $X$ is the total amount shown and $Y$ is the number of heads.

From a bag containing four balls numbered $1,2,3$, and 4 you draw two. If at least one of the numbers drawn is greater than 2 , you win $\$ 1$, and otherwise you lose $\$ 1$. Let $X$ be the amount won or lost and let $Y$ be the first number drawn.

Let $\Omega =\{ 1,2, \ldots \}$ be equipped with the $\sigma$-field of all subsets of $\Omega$ and a probability measure $P$ such that $\mathbb {P}\left(\{ k\} \right)=2^{-k}$ for each $k=1,2, \ldots$. Find $\mathbb {E}\left[ X \mid Y \right]$ if $X(k)=k$ and $Y(k)=(-1)^{k}$ for each $k=1,2, \ldots$.

A coin is tossed three times behind a curtain and you win an amount $X$ equal to the number of heads shown. If, peeping behind the curtain, you could see the result $Y$ of the first toss, how would you assess your expected winnings?

A bag contains three coins, but one has heads on both sides. Two coins are drawn and tossed. Find $\mathbb {E}\left[ X \mid Y \right]$ if $X$ is the number of heads and $Y$ is the number of genuine coins among the drawn ones.

Show that if $X$ and $Y$ are independent random variables and $Y$ is discrete, then $\mathbb {E}\left[ X \mid Y \right]=\mathbb {E}\left[X\right]$.

Show that $\mathcal{F}_{Y}$ is the smallest $\sigma$-field with respect to which $Y$ is a random variable.

A die is rolled twice; $Y$ is the first outcome and $X$ is the sum of outcomes.

Three coins, $5 \phi , 10 \phi$, and $25 \phi$, are tossed and you can take those showing heads; $X$ is the total amount you take and $Y$ is the number of heads.

From a bag containing four balls numbered $1,2,3$, and 4 , you draw two; if at least one of the numbers drawn is greater than 2 , you win $\$ 1$, and otherwise you lose $\$ 1$. Let $X$ be the amount won or lost and let $Y$ be the first number drawn.

A die is rolled twice; if the sum of outcomes is even, you win $\$ 1$, and if it is odd, you lose $\$ 1 ; X$ is the amount won or lost and $Y$ is the outcome of the first roll.

A die is rolled twice; $Y$ is the outcome of the first roll and $X$ is the outcome of the second roll.

Let $Y$ be a discrete random variable. Show that for $\mathbb {E}\left[ X \mid Y \right]$ given by Definition 10.3,

Let $Y$ be a discrete random variable. Show that

Let $Y$ be a discrete random variable. For any $B \in \mathcal{F}_{Y}$, prove that

$X(\omega )=2 \omega , Y(\omega )=2$.

$X(\omega )=2 \omega$ for all $\omega \in [0,1] ; Y(\omega )=0$ if $\omega \in \left[0, \frac{1}{2}\right)$ and $Y(\omega )=1$ if $\omega \in \left[\frac{1}{2}, 1\right]$.

$X(\omega )=2 \omega$ and $Y(\omega )=\omega^{2}$.

$X(\omega )=2 \omega ; Y(\omega )=\omega$ if $\omega \in \left[0, \frac{1}{2}\right)$ and $Y(\omega )=\frac{1}{2}$ if $\omega \in \left[\frac{1}{2}, 1\right]$.

$X=2 \omega -1+\left|2 \omega -1\right|$ and $Y=1-\left|2 \omega -1\right|$.

$X(\omega )=1-\omega ; Y(\omega )=0$ if $\omega \in \left[0, \frac{1}{2}\right)$ and $Y(\omega )=1$ if $\omega \in \left[\frac{1}{2}, 1\right]$.

$X(\omega )=1-\omega ; Y(\omega )=0$ if $\omega \in \left[0, \frac{1}{2}\right)$ and $Y(\omega )=\omega$ if $\omega \in \left[\frac{1}{2}, 1\right]$.

Show that for any $X$ and $Y$,

Prove that for any random variables $X, Y$, and $Z$ such that $\mathcal{F}_{Y} \subset \mathcal{F}_{Z}$,

Prove that if $X$ and $Y$ are independent random variables, then

Assuming that $\mathcal{F}_{Z} \subset \mathcal{F}_{Y}$, prove that

Prove that

Show without referring to Problem 10.38 that $\mathbb {E}\left[ X \mid Y \right]=\mathbb {E}\left[X\right]$ if $X$ and $Y$ are jointly continuous and independent.

Find $\mathbb {E}\left[ X \mid Y \right]$ if the joint density of $X$ and $Y$ is of the following form with a constant $K$ such that $\int_{-\infty }^{+\infty } \int_{-\infty }^{+\infty } f_{X, Y}(x, y) \mathrm{d} x \mathrm{~ d} y=1$.

$f_{X, Y}(x, y)=K(x+y)$ for $0 \leqslant x, y \leqslant 1$ and zero otherwise.

$f_{X, Y}(x, y)=K \cos x \cos y$ for $0 \leqslant x, y \leqslant \frac{\pi }{2}$ and zero otherwise.

$f_{X, Y}(x, y)=K(2 x+y)$ for $x^{2}+y^{2} \leqslant 1$ and zero otherwise.

$f_{X, Y}(x, y)=K(x+y)^{2}$ for $0 \leqslant x, y \leqslant 1$ and zero otherwise.

$f_{X, Y}(x, y)=K\left(x+y^{2}\right)$ for $x^{2}+y^{2} \leqslant 1$ and zero otherwise.

In a game played on a 3 -square board a coin is tossed repeatedly: heads allow the player to move to the next square and tails to jump over the next square. If the end of the board is reached, the reward is $\$ 10$. However, there is a trap on one of the squares, placed randomly with uniform probability. If the player falls into the trap, the game stops and he has to pay a penalty of $\$ 10$. Let $X$ be the gain or loss made by the player and let $Y$ be the number of the square containing the trap. Compute $\mathbb {E}\left[ X \mid Y \right]$ and $\mathbb {E}\left[X\right]$.

We buy $Y$ raffle tickets, where $Y$ has the Poisson distribution with parameter $\lambda$. Each ticket wins a prize with probability $p$ independently of the others. Find the conditional expectation of the number of prizes won $X$ given the number of tickets bought $Y$, and compute the expectation of $X$.

At a given moment, there are $N$ trucks on a bridge. Let $Y_{i}$ be the weight of the $i$ th truck and let $X$ be the total weight of the trucks. Show that if $N$ and all the $Y_{i}$ are independent, $\mathbb {E}\left[Y_{i}\right]=m$, and $\mathbb {E}\left[N\right]=n$, then $\mathbb {E}\left[X\right]=m n$.

The number of mistakes made by a typesetter on any given page is a random variable having the Poisson distribution with parameter $\lambda$, independent for each page. A three-page article is prepared by one of four typesetters, for whom the values of $\lambda$ are $1,2,3$, and 4 , respectively. The typesetter is selected randomly with uniform probability. What is the expected number of misprints throughout the article?

Suppose that the number of customers in a shop on a given day has the Poisson distribution with parameter $\lambda$. Each customer makes a purchase with probability $p$ independently of other customers. What is the expected number of customers who will make a purchase today?

The number of customers visiting a shop on a given day has the Poisson distribution with parameter $\lambda$. The expected number of items bought by each customer is 2 , the purchases of each customer being independent. What is the expected number of items the shop will sell in a day?