Random Variables

 Let$X$ be the winnings of a gambler. Let $p\left(i\right)=$$P(X=i)$ and suppose that

$p\left(0\right)=1 / 3 ; p\left(1\right)=p(-1)=13 / 55$

$p\left(2\right)=p(-2)=1 / 11 ; p\left(3\right)=p(-3)=1 / 165$

Compute the conditional probability that the gambler wins $i, i=1,2,3,$ given that he wins a positive amount.

In the game of Two-Finger Morra, $2$ players show $1$ or $2$ fingers and simultaneously guess the number of fingers their opponent will show. If only one of the players guesses correctly, he wins an amount (in dollars) equal to the sum of the fingers shown by him and his opponent. If both players guess correctly or if neither guesses correctly, then no money is exchanged. Consider a specified player, and denote by X the amount of money he wins in a single game of Two-Finger Morra.

 (a) If each player acts independently of the other, and if each player makes his choice of the number of fingers he will hold up and the number he will guess that his opponent will hold up in such a way that each of the $4$ possibilities is equally likely, what are the possible values of $X$ and what are their associated probabilities?

(b) Suppose that each player acts independently of the other. If each player decides to hold up the same number of fingers that he guesses his opponent will hold up, and if each player is equally likely to hold up $1$or $2$ fingers, what are the possible values of$X$ and their associated probabilities? 

Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. What are the possible values of X?

In Problem $4.5,$for $n=3,$if the coin is assumed fair, what are the probabilities associated with the values that X can take on? 

A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability $.3,$ and his second will lead independently to a sale with probability $.6$. Any sale made is equally likely to be either for the deluxe model, which costs $\backslash (1000$, or the standard model, which costs $\backslash )500.$ Determine the probability mass function of $X$, the total dollar value of all sales 

Five men and $5$ women are ranked according to their scores on an examination. Assume that no two scores are alike and all $10!$ possible rankings are equally likely. Let$X$denote the highest ranking achieved by a woman. (For instance,$X = 1$ if the top-ranked person is female.) Find$P\{X=i\},i=1,2,3,\dots ,8,9,10$

Repeat Example $1C$ when the balls are selected with replacement. 

Suppose that a die is rolled twice. What are the possible values that the following random variables can take on: 

(a) the maximum value to appear in the two rolls; 

(b) the minimum value to appear in the two rolls; 

(c) the sum of the two rolls; 

(d) the value of the first roll minus the value of the second roll? 

Two fair dice are rolled. Let $X$ equal the product of the $2$ dice. Compute $P\{X=i\}\text{ for }i=1,\dots ,36$.

Three dice are rolled. By assuming that each of the $6^3=216$ possible outcomes is equally likely, find the probabilities attached to the possible values that X can take on, where X is the sum of the 3 dice.

A certain typing agency employs $2$ typists. The average number of errors per article is $3$ when typed by the first typist and $4.2$ when typed by the second. If your article is equally likely to be typed by either typist, approximate the probability that it will have no errors.  

(a) An integer $N$ is to be selected at random from $\left\{1,2,\dots ,(10{)}^3\right\}$ in the sense that each integer has the same probability of being selected. What is the probability that $N$ will be divisible by $3$ ? by $5$ ? by $7$ ? by $15$ ? by $105$ ? How would your answer change if $(10{)}^3$ is replaced by $(10{)}^k$ as $k$ became larger and larger?

(b) An important function in number theory-one whose properties can be shown to be related to what is probably the most important unsolved problem of mathematics, the Riemann hypothesis - is the Möbius function $\mu \left(n\right)$, defined for all positive integral values $n$ as follows: Factor $n$ into its prime factors. If there is a repeated prime factor, as in $12=2·2·3$ or $49=7·7$, then $\mu \left(n\right)$ is defined to equal $0$ . Now let $N$ be chosen at random from $\left\{1,2,\dots (10{)}^k\right\}$, where $k$ is large. Determine $P\left\{\mu \right(N)=0\}$ as $k\to \infty$. Hint: To compute $P\left\{\mu \right(N)\ne 0\}$, use the identity

$\prod _{i=1}^{\infty }\frac{P_i^2-1}{P_i^2}=\left(\frac{3}{4}\right)\left(\frac{8}{9}\right)\left(\frac{24}{25}\right)\left(\frac{48}{49}\right)\cdots =\frac{6}{{\pi }^2}$

where $P_i$ is the $i$ th-smallest prime. (The number $1$ is not a prime.)

If the die in Problem 4.7 is assumed fair, calculate the probabilities associated with the random variables in parts (a) through (d). 

Four buses carrying 148 students from the same school arrive at a football stadium. The buses carry, respectively, 40, 33, 25, and 50 students. One of the students is randomly selected. Let X denote the number of students who were on the bus carrying the randomly selected student. One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on her bus.

 (a) Which of E[X] or E[Y] do you think is larger? Why? 

(b) Compute E[X] and E[Y]. 

Five distinct numbers are randomly distributed to players numbered$1$through $5.$ Whenever two players compare their numbers, the one with the higher one is declared the winner. Initially, players$1$and $2$ compare their numbers; the winner then compares her number with that of player $3,$ and so on. Let $X$ denote the number of times player $1$ is a winner. Find $PX=i,i=0,1,2,3,4.$

The National Basketball Association (NBA) draft lottery involves the 11 teams that had the worst won-lost records during the year. A total of 66 balls are placed in an urn. Each of these balls is inscribed with the name of a team: Eleven have the name of the team with the worst record, 10 have the name of the team with the second worst record, 9 have the name of the team with the third worst record, and so on (with 1 ball having the name of the team with the 11 th-worst record). A ball is then chosen at random, and the team whose name is on the ball is given the first pick in the draft of players about to enter the league. Another ball is then chosen, and if it "belongs" to a team different from the one that received the first draft pick, then the team to which it belongs receives the second draft pick. (If the ball belongs to the team receiving the first pick, then it is discarded and another one is chosen; this continues until the ball of another team is chosen.) Finally, another ball is chosen, and the team named on the ball (provided that it is different from the previous two teams) receives the third draft pick. The remaining draft picks 4 through 11 are then awarded to the 8 teams that did not "win the lottery," in inverse order of their won-lost not receive any of the 3 lottery picks, then that team would receive the fourth draft pick. Let X denote the draft pick of the team with the worst record. Find the probability mass function of X.

Suppose that the distribution function of X given by

$F\left(b\right)=\left\{\begin{array}{l}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}b<0\\ \frac{b}{4}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\le b<1\\ \frac{1}{2}+\frac{b-1}{4}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\le b<2\\ \frac{11}{12}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\le b<3\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\le b\end{array}\right.$

(a) Find $P\{X=i\},i=1,2,3$.

(b) Find $P\left\{\frac{1}{2}<X<\frac{3}{2}\right\}$.

 In Problem $4.15$, let team number $1$ be the team with the worst record, let team number $2$ be the team with the second-worst record, and so on. Let $Y_i$ denote the team that gets the draft pick number $i$. (Thus, $Y_1=3$ if the first ball chosen belongs to team number $3$.) Find the probability mass function of

 (a) $Y_1$   

(b) $Y_2$

 (c) $Y_3$.

Four independent flips of a fair coin are made. Let $X$ denote the number of heads obtained. Plot the probability mass function of the random variable $X-2$. 

4.20. A gambling book recommends the following "winning strategy" for the game of roulette: Bet $\backslash (1$ on red. If red appears (which has probability $\frac{18}{38}$), then take the $\backslash )1$ profit and quit. If red does not appear and you lose this bet (which has probability $\frac{20}{38}$ of occurring), make additional $\$1$ bets on red on each of the next two spins of the roulette wheel and then quit. Let $X$ denote your winnings when you quit.

(a) Find $P\{X>0\}$.

(b) Are you convinced that the strategy is indeed a "winning" strategy? Explain your answer!

(c) Find $E\left[X\right]$.

If the distribution function of $X$ is given by

$F\left(b\right)=\left\{\begin{array}{l}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}b<0\\ \frac{1}{2}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\le b<1\\ \frac{3}{5}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\le b<2\\ \frac{4}{5}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\le b<3\\ \frac{9}{10}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}3\le b<3.5\\ 1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}b\ge 3.5\end{array}\right.$

calculate the probability mass function of $X$. 

Suppose that two teams play a series of games that ends when one of them has won $i$ games. Suppose that each game played is, independently, won by team $A$ with probability $p$. Find the expected number of games that are played when

 (a) $i=2$ and

(b) $i=3$. Also, show in both cases that this number is maximized when $p=\frac{1}{2}$.

You have $\backslash (1000$, and a certain commodity presently sells for $\backslash )2$ per ounce. Suppose that after one week the commodity will sell for either $\backslash (1$or $\backslash )4$ an ounce, with these two possibilities being equally likely.

(a) If your objective is to maximize the expected amount of money that you possess at the end of the week, what strategy should you employ?

(b) If your objective is to maximize the expected amount of the commodity that you possess at the end of the week, what strategy should you employ? 

A sample of 3 items is selected at random from a box containing 20 items of which 4 are defective. Find the expected number of defective items in the sample. 

There are two possible causes for a breakdown of a machine. To check the first possibility would cost C1 dollars, and, if that were the cause of the breakdown, the trouble could be repaired at a cost of R1 dollars. Similarly, there are costs C2 and R2 associated with the second possibility. Let p and 1 − p denote, respectively, the probabilities that the breakdown is caused by the first and second possibilities. Under what conditions on p, Ci, Ri, i = 1, 2, should we check the first possible cause of breakdown and then the second, as opposed to reversing the checking order, so as to minimize the expected cost involved in returning the machine to working order? 

 Two coins are to be flipped. The first coin will land on heads with probability .$6$, the second with probability .$7$. Assume that the results of the flips are independent, and let X equal the total number of heads that result. (a) Find P{X =$1$}. (b) Determine E[X].

One of the numbers $1$ through $10$ is randomly chosen. You are to try to guess the number chosen by asking questions with “yes-no” answers. Compute the expected number of questions you will need to ask in each of the following two cases:

(a) Your ith question is to be “Is it i?” i = $1,$ $2,$ $3,$ $4,$ $5,$ $6,$ $7,$ $8,$ $9,$ $10$. (b) With each question, you try to eliminate one-half of the remaining numbers, as nearly as possible.

Each night different meteorologists give us the probability that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability $p$, then he or she will receive a score of

$1-(1-p{)}^2\text{ if it does rain }$

$1-p^2 if it does not rain$

We will then keep track of scores over a certain time span and conclude that the meteorologist with the highest average score is the best predictor of weather. Suppose now that a given meteorologist is aware of our scoring mechanism and wants to maximize his or her expected score. If this person truly believes that it will rain tomorrow with probability $p^{*}$, what value of $p$ should he or she assert so as to maximize the expected score?

To determine whether they have a certain disease, $100$ people are to have their blood tested. However, rather than testing each individual separately, it has been decided first to place the people into groups of $10$. The blood samples of the $10$ people in each group will be pooled and analyzed together. If the test is negative, one test will suffice for the $10$ people, whereas if the test is positive, each of the $10$ people will also be individually tested and, in all, $11$ tests will be made on this group. Assume that the probability that a person has the disease is  $.1$ for all people, independently of one another, and compute the expected number of tests necessary for each group. (Note that we are assuming that the pooled test will be positive if at least one person in the pool has the disease.)

A person tosses a fair coin until a tail appears for the first time. If the tail appears on the n th flip, the person wins 2n dollars. Let X denote the player's winnings. Show that $E\left[X\right]=+\infty$. This problem is known as the St. Petersburg paradox.

(a) Would you be willing to pay \( 1 million to play this game once?

(b) Would you be willing to pay \) 1 million for each game if you could play for as long as you liked and only had to settle up when you stopped playing?

An insurance company writes a policy to the effect that an amount of money $A$ must be paid if some event $E$ occurs within a year. If the company estimates that $E$ will occur within a year with probability $p$, what should it charge the customer in order that its expected profit will be 10 percent of $A$ ?

$A$ and $B$ play the following game: $A$ writes down either number $1$ or number $2$ , and $B$ must guess which one. If the number that $A$ has written down is $i$ and $B$ has guessed correctly, $B$ receives $i$ units from $A$. If $B$ makes a wrong guess, $B$ pays $\frac{3}{4}$ unit to $A$. If $B$ randomizes his decision by guessing 1 with probability $p$ and $2$ with probability $1-p$, determine his expected gain if (a) $A$ has written down number $1$ and (b) $A$ has written down number $2$ .

What value of $p$ maximizes the minimum possible value of $B$ 's expected gain, and what is this maximin value? (Note that $B$ 's expected gain depends not only on $p$, but also on what $A$ does.)

Consider now player $A$. Suppose that she also randomizes her decision, writing down number $1$ with probability $q$. What is $A$ 's expected loss if (c) $B$ chooses number 1 and (d) $B$ chooses number $2$ ?

What value of $q$ minimizes $A$ 's maximum expected loss? Show that the minimum of $A$ 's maximum expected loss is equal to the maximum of $B$ 's minimum expected gain. This result, known as the minimax theorem, was first established in generality by the mathematician John von Neumann and is the fundamental result in the mathematical discipline known as the theory of games. The common value is called the value of the game to player $B$.

Consider Problem 4.22 with i = 2. Find the variance of the number of games played, and show that this number is maximized when p = 1 2 . 

Find Var(X) and Var(Y) for X and Y as given in Problem 4.21 

If E[X] = 1 and Var(X) = 5, find 

(a) E[(2 + X)2]; 

(b) Var(4 + 3X). 

Suppose that it takes at least $9$votes from a $12$- member jury to convict a defendant. Suppose also that the probability that a juror votes a guilty person innocent is $.2,$ whereas the probability that the juror votes an innocent person guilty is $.1.$ If each juror acts independently and if $65$ percent of the defendants are guilty, find the probability that the jury renders a correct decision. What percentage of defendants is convicted? 

In some military courts, $9$ judges are appointed. However, both the prosecution and the defense attorneys are entitled to a peremptory challenge of any judge, in which case that judge is removed from the case and is not replaced. A defendant is declared guilty if the majority of judges cast votes of guilty, and he or she is declared innocent otherwise. Suppose that when the defendant is, in fact, guilty, each judge will (independently) vote guilty with probability .$7,$whereas when the defendant is, in fact, innocent, this probability drops to .$3.$ 

(a) What is the probability that a guilty defendant is declared guilty when there are (i) $9$, (ii) $8$, and (iii) $7$judges?

(b) Repeat part (a) for an innocent defendant. 

(c) If the prosecuting attorney does not exercise the right to a peremptory challenge of a judge, and if the defense is limited to at most two such challenges, how many challenges should the defense attorney make if he or she is $60$ percent certain that the client is guilty? 

A newsboy purchases papers at $10$ cents and sells them at $15$ cents. However, he is not allowed to return unsold papers. If his daily demand is a binomial random variable with $n=10,p=\frac{1}{3}$, approximately how many papers should he purchase so as to maximize his expected profit?

In Example $4b$, suppose that the department store incurs an additional cost of $c$ for each unit of unmet demand. (This type of cost is often referred to as a goodwill cost because the store loses the goodwill of those customers whose demands it cannot meet.) Compute the expected profit when the store stocks $s$ units, and determine the value of data-custom-editor="chemistry" $\mathrm{s}$ that maximizes the expected profit.

A box contains $5$ red and $5$ blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win $\backslash (1.10$; if they are different colors, then you win $-\backslash )1.00$. (That is, you lose $\$1.00$.) Calculate

(a) the expected value of the amount you win;

(b) the variance of the amount you win.

A ball is drawn from an urn containing $3$ white and $3$ black balls. After the ball is drawn, it is replaced and another ball is drawn. This process goes on indefinitely. What is the probability that the first $4$ balls drawn, exactly $2$ are white?

On a multiple-choice exam with $3$possible answers for each of the $5$ questions, what is the probability that a student will get $4$ or more correct answers just by guessing? 

A man claims to have extrasensory perception. As a test, a fair coin is flipped $10$ times and the man is asked to predict the outcome in advance. He gets $7$out of $10$ correct. What is the probability that he would have done at least this well if he did not have ESP? 

$A$and $B$will take the same $10$-question examination. Each question will be answered correctly by $A$ with probability$.7$, independently of her results on other questions. Each question will be answered correctly by B with probability $.4$, independently both of her results on the other questions and on the performance of $A.$

(b) Find the variance of the number of questions that are answered correctly by either A or B 

A communications channel transmits the digits $0$and $1.$ However, due to static, the digit transmitted is incorrectly received with probability $2.$ Suppose that we want to transmit an important message consisting of one binary digit. To reduce the chance of error, we transmit $00000$ instead of $0$ and 11111 instead of $1.$ If the receiver of the message uses “majority” decoding, what is the probability that the message will be wrong when decoded? What independence assumptions are you making? 

A satellite system consists of$n$ components and functions on any given day if at least $k$ of the n components function on that day. On a rainy day, each of the components independently functions with probability $p1,$ whereas, on a dry day, each independently functions with probability $p2$. If the probability of rain tomorrow is what is the probability that the satellite system will function? 

A student is getting ready to take an important oral examination and is concerned about the possibility of having an “on” day or an “off” day. He figures that if he has an on the day, then each of his examiners will pass him, independently of one another, with probability$8$, whereas if he has an off day, this probability will be reduced to$4$. Suppose that the student will pass the examination if a majority of the examiners pass him. If the student believes that he is twice as likely to have an off day as he is to have an on the day, should he request an examination with$3$examiners or with$5$examiners? 

Suppose that a biased coin that lands on heads with probability $p$ is flipped $10$ times. Given that a total of $6$ heads results, find the conditional probability that the first $3$ outcomes are

(a) $h, t, t$ (meaning that the first flip results in heads, the second is tails, and the third in tails);

(b) $t, h, t.$

How many people are needed so that the probability that at least one of them has the same birthday as you is greater than $\frac{1}{2}$ ? 

Suppose that the number of accidents occurring on a highway each day is a Poisson random variable with parameter λ = 3.

(a) Find the probability that 3 or more accidents occur today. 

(b) Repeat part (a) under the assumption that at least 1 accident occurs today.