In Example 2.2-1 let Z = u(X) = X3. (a) Find the pmf of Z, say h(z). (b) Find E(Z). (c)

Let the random variable \(X\) have the pmf \(f(x)=\frac{(|x|+1)^{2}}{9}, \quad x=-1,0,1\). Compute \9E(X), E\left(X^{2}\right)\), and \(E\left(3 X^{2}-2 X+4\right)\). Equation Transcription: Text Transcription: X f(x)=(|x|+1)^2/9, x=-1,0,1 E(X),E(X^2) E(3X^2-2X+4)

Find \(E(X)\) for each of the distributions given in Exercise 2.1-3. Equation Transcription: Text Transcription: E(X)

An insurance company sells an automobile policy with a deductible of one unit. Let \(X\) be the amount of the loss having pmf \(f(x)= \begin{cases}0.9, & x=0 \\ \frac{c}{x}, & x=1,2,3,4,5,6\end{cases}\) where \(c\) is a constant. Determine \(c\) and the expected value of the amount the insurance company must pay. Equation Transcription: { Text Transcription: X c f(x)= { _c/x, x=1,2,3,4,5,6 ^0.9 x=0

Let the random variable \(X\) be the number of days that a certain patient needs to be in the hospital. Suppose \(X\) has the pmf \(f(x)=\frac{5-x}{10}, \quad x=1,2,3,4\) If the patient is to receive $200 from an insurance company for each of the first two days in the hospital and $100 for each day after the first two days, what is the expected payment for the hospitalization? Equation Transcription: Text Transcription: X f(x)=5-x/10, x=1,2,3,4

Problem 7E In the gambling game chuck-a-luck, for a $1 bet it is possible to win $1, $2, or $3 with respective probabilities 75/216, 15/216, and 1/216. One dollar is lost with probability 125/216. Let X equal the payoff for this game and find E(X). Note that when a bet is won, the $1 that was bet, in addition to the $1, $2, or $3 that is won, is returned to the bettor.

Problem 8E Let X be a random variable with support {1, 2, 3, 5, 15, 25, 50}, each point of which has the same probability 1/7. Argue that c = 5 is the value that minimizes h(c) = E( |X ? c| ). Compare c with the value of b that minimizes g(b) = E[(X ? b)2].

Problem 12E Suppose that a school has 20 classes: 16 with 25 students in each, three with 100 students in each, and one with 300 students, for a total of 1000 students. (a) What is the average class size? (b) Select a student randomly out of the 1000 students. Let the random variable X equal the size of the class to which this student belongs, and define the pmf of X. (c) Find E(X), the expected value of X. Does this answer surprise you?

Problem 11E In the gambling game craps (see Exercise 1.3- 13), the player wins $1 with probability 0.49293 and loses $1 with probability 0.50707 for each $1 bet. What is the expected value of the game to the player? References see Exercise 1.3 – 13 In the gambling game “craps,” a pair of dice is rolled and the outcome of the experiment is the sum of the points on the up sides of the six-sided dice. The bettor wins on the first roll if the sum is 7 or 11. The bettor loses on the first roll if the sum is 2, 3, or 12. If the sum is 4, 5, 6, 8, 9, or l0, that number is called the bettor’s “point.” Once the point is established, the rule is as follows: If the bettor rolls a 7 before the point, the bettor loses; but if the point is rolled before a 7, the bettor wins. (a) List the 36 outcomes in the sample space for the roll of a pair of dice. Assume that each of them has a probability of 1/36. (b) Find the probability that the bettor wins on the first roll. That is, find the probability of rolling a 7 or 11, P(7 or 11). (c) Given that 8 is the outcome on the first roll, find the probability that the bettor now rolls the point 8 before rolling a 7 and thus wins. Note that at this stage in the game the only outcomes of interest are 7 and 8. Thus find P(8 |7 or 8). (d) The probability that a bettor rolls an 8 on the first roll and then wins is given by P(8)P(8 | 7 or 8). Show that this probability is (5/36)(5/11). (e) Show that the total probability that a bettor wins in the game of craps is 0.49293. Hint: Note that the bettor can win in one of several mutually exclusive ways: by rolling a 7 or an 11 on the first roll or by establishing one of the points 4, 5, 6, 8, 9, or 10 on the first roll and then obtaining that point on successive rolls before a 7 comes up.

Find E(X) for each of the distributions given in Exercise 2.1-3.

Let the random variable X have the pmf f(x) = (|x| + 1)2 9 , x = 1, 0, 1. Compute E(X), E(X2), and E(3X2 2X + 4).

Let the random variable X be the number of days that a certain patient needs to be in the hospital. Suppose X has the pmf \(f(x)=\frac{5-x}{10}\), x = 1, 2, 3, 4. If the patient is to receive $200 from an insurance company for each of the first two days in the hospital and $100 for each day after the first two days, what is the expected payment for the hospitalization?

An insurance company sells an automobile policy with a deductible of one unit. Let X be the amount of the loss having pmf f(x) = 0.9, x = 0, c x , x = 1, 2, 3, 4, 5, 6, where c is a constant. Determine c and the expected value of the amount the insurance company must pay.

In Example 2.2-1 let Z = u(X) = X3. (a) Find the pmf of Z, say h(z). (b) Find E(Z). (c) How much, on average, can the young man expect to win on each play if he charges $10 per play?

Let the pmf of X be defined by f(x) = 6/(2x2), x = 1, 2, 3, .... Show that E(X) = + and thus, does not exist.

In the gambling game chuck-a-luck, for a $1 bet it is possible to win $1, $2, or $3 with respective probabilities 75/216, 15/216, and 1/216. One dollar is lost with probability 125/216. Let X equal the payoff for this game and find E(X). Note that when a bet is won, the $1 that was bet, in addition to the $1, $2, or $3 that is won, is returned to the bettor.

Let X be a random variable with support {1, 2, 3, 5, 15, 25, 50}, each point of which has the same probability 1/7. Argue that c = 5 is the value that minimizes h(c) = E(|X c|). Compare c with the value of b that minimizes g(b) = E[(X b)2].

A roulette wheel used in a U.S. casino has 38 slots, of which 18 are red, 18 are black, and 2 are green. A roulette wheel used in a French casino has 37 slots, of which 18 are red, 18 are black, and 1 is green. A ball is rolled around the wheel and ends up in one of the slots with equal probability. Suppose that a player bets on red. If a $1 bet is placed, the player wins $1 if the ball ends up in a red slot. (The player’s $1 bet is returned.) If the ball ends up in a black or green slot, the player loses $1. Find the expected value of this game to the player in (a) The United States. (b) France.

In the casino game called high–low, there are three possible bets. Assume that $1 is the size of the bet. A pair of fair six-sided dice is rolled and their sum is calculated. If you bet low, you win $1 if the sum of the dice {2, 3, 4, 5, 6}. If you bet high, you win $1 if the sum of the dice is {8, 9, 10, 11, 12}. If you bet on {7}, you win $4 if a sum of 7 is rolled. Otherwise, you lose on each of the three bets. In all three cases, your original dollar is returned if you win. Find the expected value of the game to the bettor for each of these three bets.

In the gambling game craps (see Exercise 1.3- 13), the player wins $1 with probability 0.49293 and loses $1 with probability 0.50707 for each $1 bet. What is the expected value of the game to the player?

Suppose that a school has 20 classes: 16 with 25 students in each, three with 100 students in each, and one with 300 students, for a total of 1000 students. (a) What is the average class size? (b) Select a student randomly out of the 1000 students. Let the random variable X equal the size of the class to which this student belongs, and define the pmf of X. (c) Find E(X), the expected value of X. Does this answer surprise you?