Properties of Expectation

A fair die is rolled $10$ times. Calculate the expected sum of the$10$ rolls. 

Consider 3 trials, each having the same probability of success. Let $X$ denote the total number of successes in these trials. If $E\left[X\right] = 1.8$, what is

(a) the largest possible value of $PX=3$? 

(b) the smallest possible value of $P\{X=3\}$}?  

The county hospital is located at the center of a square whose sides are $3$ miles wide. If an accident occurs within this square, then the hospital sends out an ambulance. The road network is rectangular, so the travel distance from the hospital, whose coordinates are $(0,0)$, to the point $(x, y)$ is $\left|x\right|+\left|y\right|$. If an accident occurs at a point that is uniformly distributed in the square, find the expected travel distance of the ambulance. 

The game of Clue involves 6 suspects, 6 weapons, and 9 rooms. One of each is randomly chosen and the object of the game is to guess the chosen three.

 (a) How many solutions are possible? In one version of the game, the selection is made and then each of the players is randomly given three of the remaining cards. Let S, W, and R be, respectively, the numbers of suspects, weapons, and rooms in the set of three cards given to a specified player. Also, let X denote the number of solutions that are possible after that player observes his or her three cards. 

(b) Express X in terms of S, W, and R. 

(c) Find E[X]

Gambles are independent, and each one results in the player being equally likely to win or lose 1 unit. Let W denote the net winnings of a gambler whose strategy is to stop gambling immediately after his first win. Find

 (a) P{W > 0}

 (b) P{W < 0} 

 (c) E[W] 

7.4. If X and Y have joint density function $f_{X,Y}(x,y)=\{\begin{array}{ll}1/y,& \text{if }0<y<1,0<x<y\\ 0,& \text{otherwise}\end{array}$ find

(a) E[X Y]

(b) E[X]

(c) E[Y]

Suppose that A and B each randomly and independently choose$3$ of $10$objects. Find the expected number of objects

a. Chosen by both A and B; 

b. Not chosen by either A or B; 

c. Chosen by exactly one of A and B.  

N people arrive separately to a professional dinner. Upon arrival, each person looks to see if he or she has any friends among those present. That person then sits either at the table of a friend or at an unoccupied table if none of those present is a friend. Assuming that each of the $\left(\begin{array}{l}N\\ 2\end{array}\right)$ pairs of people is, independently, a pair of friends with probability p, find the expected number of occupied tables.

Hint: Let $X_i$ equal $1$ or $0$, depending on whether the $i$th arrival sits at a previously unoccupied table. 

A total of n balls, numbered $1$ through n, are put into n urns, also numbered $1$ through $n$ in such a way that ball $i$ is equally likely to go into any of the urns $1,2,..i$ .

Find (a) the expected number of urns that are empty.

(b) the probability that none of the urns is empty. 

Consider $n$ independent flips of a coin having probability $p$ of landing on heads. Say that a changeover occurs whenever an outcome differs from the one preceding it. For instance, if $n = 5$ and the outcome is$HHTHT$, then there are $3$ changeovers. Find the expected number of changeovers. Hint: Express the number of changeovers as the sum of $n - 1$ Bernoulli random variables. 

A set of $1000$ cards numbered 1 through $1000$ is randomly distributed among $1000$ people with each receiving one card. Compute the expected number of cards that are given to people whose age matches the number on the card. 

In Example $2$h,say that i and$j,i\ne j$ , form a matched pair if i chooses the hat belonging to j and j chooses the hat belonging to i. Find the expected number of matched pairs.

 Let Z be a standard normal random variable,and, for a fixed x, set 

$X=\{\begin{array}{ll}Z& \text{if}Z>x\\ 0& \text{otherwise}\end{array}$

$\text{Show that}E\left[X\right]=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}\text{.}$

 A deck of n cards numbered 1 through n is thoroughly shuffled so that all possible n! orderings can be assumed to be equally likely. Suppose you are to make n guesses sequentially, where the ith one is a guess of the card in position i. Let N denote the number of correct guesses. 

(a) If you are not given any information about your earlier guesses, show that for any strategy, E[N]=1. 

(b) Suppose that after each guess you are shown the card that was in the position in question. What do you think is the best strategy? Show that under this strategy

$\begin{array}{rl}E\left[N\right]& =\frac{1}{n}+\frac{1}{n-1}+\cdots +1\\ & \approx {\int }_1^n\frac{1}{x}dx=\log n\end{array}$

(c) Supposethatyouaretoldaftereachguesswhetheryou are right or wrong. In this case, it can be shown that the strategy that maximizes E[N] is one that keeps on guessing the same card until you are told you are correct and then changes to a new card. For this strategy, show that 

$\begin{array}{rl}E\left[N\right]& =1+\frac{1}{2!}+\frac{1}{3!}+\cdots +\frac{1}{n!}\\ & \approx e-1\end{array}$

Hint: For all parts, express N as the sum of indicator (that is, Bernoulli) random variables. 

Cards from an ordinary deck of $52$ playing cards are turned face upon at a time. If the 1st card is an ace, or the $2$nd a deuce, or the $3$rd a three, or ...,or the $13$th a king,or the $14$ an ace, and so on, we say that a match occurs. Note that we do not require that the ($13$n + $1$) card be any particular ace for a match to occur but only that it be an ace. Compute the expected number of matches that occur.

A certain region is inhabited by r distinct types of a certain species of insect. Each insect caught will, independently of the types of the previous catches, be of type i with probability 

$P_i,i=1,\dots ,r\sum _1^r{P}_i=1$

(a) Compute the mean number of insects that are caught before the first type $1$ catch.

(b) Compute the mean number of types of insects that are caught before the first type $1$ catch.

An urn has $m$ black balls. At each stage, a black ball is removed and a new ball that is black with probability $p$ and white with probability $1 - p$ is put in its place. Find the expected number of stages needed until there are no more black balls in the urn. note: The preceding has possible applications to understanding the AIDS disease. Part of the body’s immune system consists of a certain class of cells known as T-cells. There are$2$ types of T-cells, called CD4 and CD8. Now, while the total number of T-cells in AIDS sufferers is (at least in the early stages of the disease) the same as that in healthy individuals, it has recently been discovered that the mix of CD4 and CD8 T-cells is different. Roughly 60 percent of the T-cells of a healthy person are of the CD4 type, whereas the percentage of the T-cells that are of CD4 type appears to decrease continually in AIDS sufferers. A recent model proposes that the HIV virus (the virus that causes AIDS) attacks CD4 cells and that the body’s mechanism for replacing killed T-cells does not differentiate between whether the killed T-cell was CD4 or CD8. Instead, it just produces a new T-cell that is CD4 with probability .$6$ and CD8 with probability .$4$. However, although this would seem to be a very efficient way of replacing killed T-cells when each one killed is equally likely to be any of the body’s T-cells (and thus has probability .$6$ of being CD4), it has dangerous consequences when facing a virus that targets only the CD4 T-cells 

A group of $n$ men and $n$ women is lined up at random. 

(a) Find the expected number of men who have a woman next to them. 

(b) Repeat part (a), but now assuming that the group is randomly seated at a round table. 

In Problem 7.6, calculate the variance of the sum of the rolls.

In Problem 7.9, compute the variance of the number of empty urns.

If $E\left[X\right]=1$and $Var\left(X\right)=5$ find

(a) $E\left[\right(2+X^2\left)\right]$

(b) $Var(4+3 X)$

If 10 married couples are randomly seated at a round table, compute 

(a) The expected number and 

(b) The variance of the number of wives who are seated next to their husbands.

Cards from an ordinary deck are turned face up one at a time. Compute the expected number of cards that need to be turned face up in order to obtain

(a) 2 aces;

(b) 5 spades;

(c) all 13 hearts.

 In an urn containing n balls, the ith ball has weight W(i),i = $1$,...,n. The balls are removed without replacement, one at a time, according to the following rule: At each selection, the probability that a given ball in the urn is chosen is equal to its weight divided by the sum of the weights remaining in the urn. For instance, if at some time i$1$,...,ir is the set of balls remaining in the urn, then the next selection will be ij with probability $W\left(i_j\right)/\sum _{k=1}^{r}W\left(i_k\right)$, j = 1,...,r  Compute the expected number of balls that are withdrawn before the ball number $1$ is removed.

For a group of 100 people, compute 

(a) the expected number of days of the year that are birthdays of exactly 3 people;

(b) the expected number of distinct birthdays. 

How many times would you expect to roll a fair die before all $6$ sides appeared at least once? 

Urn $1$ contains $5$ white and $6$ black balls, while urn $2$ contains $8$ white and $10$ black balls. Two balls are randomly selected from urn $1$ and are put into urn $2$. If $3$ balls are then randomly selected from urn $2$, compute the expected number of white balls in the trio.

 Hint: LetXi = $1$ if the i th white ball initially in urn $1$ is one of the three selected, and let Xi = $0$ otherwise. Similarly, let Yi = $1$ if the i the white ball from urn $2$ is one of the three selected, and let Yi = $0$ otherwise. The number of white balls in the trio can now be written as $\sum _1^5{X}_i+\sum _1^8{Y}_i$

A bottle initially contains m large pills and n small pills. Each day, a patient randomly chooses one of the pills. If a small pill is chosen, then that pill is eaten. If a large pill is chosen, then the pill is broken in two; one part is returned to the bottle (and is now considered a small pill) and the other part is then eaten.

 (a) Let X denote the number of small pills in the bottle after the last large pill has been chosen and its smaller half returned. Find E[X].

 Hint: Define n + m indicator variables, one for each of the small pills initially present and one for each of the small pills created when a large one is split in two. Now use the argument of Example $2$m.

(b) Let Y denote the day on which the last large pills chosen. Find E[Y]. 

Hint: What is the relationship between X and Y? 

There are $4$ different types of coupons, the first $2$ of which comprise one group and the second $2$ another group. Each new coupon obtained is type i with probability $pi$, where $p1 = p2 = 1/8, p3 = p4 = 3/8$. Find the expected number of coupons that one must obtain to have at least one of

(a) all $4$ types;

(b) all the types of the first group; 

(c) all the types of the second group; 

(d) all the types of either group 

If $X$ and Y are independent and identically distributed with mean $\mu$ and variance $\sigma 2$, find 

$E\left[(X-Y{)}^2\right]$

Let $X_1,X_2,\dots$ be a sequence of independent and identically distributed continuous random variables. Let $N\ge 2$ be such that

$X_1\ge X_2\ge \cdots \ge X_{N-1}<X_N$

That is, $N$ is the point at which the sequence stops decreasing. Show that $E\left[N\right]=e$.

Hint: First find $P\{N\ge n\}$.

If $X_1,X_2,\dots ,X_n$ are independent and identically distributed random variables having uniform distributions over $(0,1)$, find

(a) $E\left[\max \left(X_1,\dots ,X_n\right)\right]$;

(b) $E\left[\min \left(X_1,\dots ,X_n\right)\right]$.

If $101$ items are distributed among $10$ boxes, then at least one of the boxes must contain more than $10$ items. Use the probabilistic method to prove this result. 

The $k$-of-$r$-out-of- $n$ circular reliability system, $k\le r\le n$, consists of $n$ components that are arranged in a circular fashion. Each component is either functional or failed, and the system functions if there is no block of $r$ consecutive components of which at least $k$ are failed. Show that there is no way to arrange $47$ components, $8$ of which are failed, to make a functional $3$-of-$12$-out-of- $47$ circular system.

Consider the following dice game, as played at a certain gambling casino: Players$1$and $2$ roll a pair of dice in turn. The bank then rolls the dice to determine the outcome according to the following rule: Player$i, i = 1, 2,$ wins if his roll is strictly greater than the banks. For$i = 1, 2,$let

$I_i=\left\{\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }i\text{ wins }\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ otherwise }\end{array}\right.$

and show that $I1$and $I2$ are positively correlated. Explain why this result was to be expected.

Let $X$ be the number of $1’s$ and $Y$ the number of $2\text{'}s$ that occur in $n$ rolls of a fair die. Compute $Cov(X, Y)$.

A die is rolled twice. Let X equal the sum of the outcomes, and let Y equal the first outcome minus the second.

Compute $Cov(X, Y).$

Consider a graph having $n$vertices labeled$1, 2, ... , n$, and suppose that, between each of the $\left(\frac{n}{2}\right)$ pairs of distinct vertices, an edge is independently present with probability $p$. The degree of a vertex$i$, designated as$Di,$ is the number of edges that have vertex $i$ as one of their vertices.

(a) What is the distribution of $Di$?

(b) Find $\rho (Di, Dj)$, the correlation between $Di$and$Dj$.

The random variables X and Y have a joint density function is given by 

$f(x,y)=\{\begin{array}{ll}2e^{-2x}/x& 0\le x<\mathrm{\infty },0\le y\le x\\ 0& \text{otherwise}\end{array}$

Compute $Cov(X,Y)$

Let $X_1, . . .$be independent with common mean $\mu$and common variance ${\sigma }_2$, and set $Y_n=X_n+X_{n+1}+X_{n+2}$. For $j\ge 0$, find $Cov\left(Y_n,Y_{n+j}\right).$

A pond contains $100$ fish, of which $30$ are carp. If $20$ fish are caught, what are the mean and variance of the number of carp among the $20$? What assumptions are you making?

The joint density function of$X$and$Y$is given by

$f(x,y)=\frac{1}{y}{e}^{-(y+x/y)}, x>0,y>0$

Find $E\left[X\right], E\left[Y\right]$ and show that $Cov(X, Y)=1$

A group of 20 people consisting of 10 men and 10 women is randomly arranged into 10 pairs of 2 each. Compute the expectation and variance of the number of pairs that consist of a man and a woman. Now suppose the 20 people consist of 10 married couples. Compute the mean and variance of the number of married couples that are paired together.

Let $X_1,X_2,\dots ,X_n$ be independent random variables having an unknown continuous distribution function $F$ and let $Y_1,Y_2,\dots ,Y_m$ be independent random variables having an unknown continuous distribution function $G$. Now order those $n+m$variables, and let

$I_i=\left\{\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if the }i\text{ th smallest of the }n+m\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ variables is from the }X\text{ sample }\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ otherwise }\end{array}\right.$

The random variable $R=\sum _{i=1}^{n+m}i{I}_i$ is the sum of the ranks of the $X$ sample and is the basis of a standard statistical procedure (called the Wilcoxon sum-of-ranks test) for testing whether $F$ and $G$ are identical distributions. This test accepts the hypothesis that $F=G$ when $R$ is neither too large nor too small. Assuming that the hypothesis of equality is in fact correct, compute the mean and variance of $R$.

Hint: Use the results of Example 3e.

Between two distinct methods for manufacturing certain goods, the quality of goods produced by method $i$ is a continuous random variable having distribution $F_i,i=1,2$. Suppose that goods are produced by method 1 and $m$ by method 2 . Rank the $n+m$ goods according to quality, and let

$X_j=\left\{\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if the }j\text{ th best was produced from }\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ method }1\\ 2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ otherwise }\end{array}\right.$

For the vector $X_1,X_2,\dots ,X_{n+m}$,  which consists of $n 1\text{'}s$ and $m 2\text{'}s$, let $R$ denote the number of runs of 1 . For instance, if $n=5, m=2$, and $X=1,2,1,1,1,1,2$, then $R=2$. If $F_1=F_2$ (that is, if the two methods produce identically distributed goods), what are the mean and variance of $R$ ?

If $X_1,X_2,X_3$, and $X_4$ are (pairwise) uncorrelated random variables, each having mean 0 and variance 1 , compute the correlations of

(a) $X_1+X_2$ and $X_2+X_3$

(b) $X_1+X_2$ and $X_3+X_4$.

A fair die is successively rolled. Let X and Y denote, respectively, the number of rolls necessary to obtain a 6 and a 5. Find

 (a) $E\left[X\right]$;

(b) $E[X\mid Y=1]$;

(c) $E[X\mid Y=5]$;

There are two misshapen coins in a box; their probabilities for landing on heads when they are flipped are, respectively, .$4$ and .$7$. One of the coins is to be randomly chosen and flipped 10 times. Given that two of the first three flips landed on heads, what is the conditional expected number of heads in the $10$ flips? 

There are $n+1$ participants in a game. Each person independently is a winner with probability $p$. The winners share a total prize of 1 unit. (For instance, if 4 people win, then each of them receives $\frac{1}{4}$ whereas if there are no winners, then none of the participants receives anything.) Let$A$denote a specified one of the players, and let $X$ denote the amount that is received by $A.$

(a) Compute the expected total prize shared by the players.

(b) Argue that $E\left[X\right]=\frac{1-(1-p{)}^{n+1}}{n+1}$

(c) Compute $E\left[X\right]$ by conditioning on whether$A$is a winner, and conclude that $E\left[(1+B{)}^{-1}\right]=\frac{1-(1-p{)}^{n+1}}{(n+1)p}$

When $B$ is a binomial random variable with parameters $n$ and $p.$

A coin having probability p of coming up heads is continually flipped until both heads and tails have appeared. Find 

(a) the expected number of flips, 

(b) the probability that the last flip lands on heads.