Questions and Answers of A First Course In Probability

Use the identity of Theoretical Exercises 5.5 to derive E[X2] when X is an exponential random variable with parameter λ.Use the result that for a nonnegative random variable Y,Theoretical Exercises
Find Var(X) and Var(Y) for X and Y as given in Problem 4.21.Problem 4.21a. Which of E[X] or E[Y] do you think is larger? Why?b. Compute E[X] and E[Y].
Repeat Theoretical Exercise 4.33, this time assuming that withdrawn chips are not replaced before the next selection.Exercise 4.33A jar contains n chips. Suppose that a boy successively draws a chip
Repeat part (a) of Problem 8.2 when it is known that the variance of a student’s test score is equal to 25.Problem 8.2a. Give an upper bound for the probability that a student’s test score will
Verify the formula for the moment generating function of a uniform random variable that is given in Table 7.2. Also, differentiate to verify the formulas for the mean and variance.Table 7.2
Repeat Problem 6.60 when X and Y are independent exponential random variables, each with parameter λ = 1.Problem 6.60If X and Y are independent and identically distributed uniform random variables
You arrive at a bus stop at 10 A.M., knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30.a. What is the probability that you will have to wait longer than 10
Each member of a population of size n is, independently, female with probability or male with probability 1 - p. Let X be the number of the other n - 1 members of the population that are the same sex
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.Problem 4.22Suppose that two teams play a series of games that
In Problem 4.5, for n = 3, if the coin is assumed fair, what are the probabilities associated with the values that can take on?Problem 4.5Let X represent the difference between the number of heads
Repeat Problem 3.87 when each of the 3 players selects from his own urn. That is, suppose that there are 3 different urns of 12 balls with 4 white balls in each urn.An urn contains 12 balls, of which
Assume, as in Example 3h , that 64 percent of twins are of the same sex. Given that a newborn set of twins is of the same sex, what is the conditional probability that the twins are identical?
Suppose that in Problem 9.2, Al is agile enough to escape from a single car, but if he encounters two or more cars while attempting to cross the road, then he is injured. What is the probability that
In a tournament involving players 1, 2, 3, 4, players 1 and 2 play a game, with the loser departing and the winner then playing against player 3, with the loser of that game departing and the winner
Consider the friendship network described by Figure 4.5. Let be a randomly chosen person and let be a randomly chosen friend of X. With f(i) equal to the number of friends of person i, show that
If E[X] = 1 and Var (X) = 5, finda. E[(2 + X)2];b. Var(4 + 3X).
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 of
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
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
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
State your assumptions. Suppose that the average number of cars abandoned weekly on a certain highway is 2.2. Approximate the probability that there will bea. No abandoned cars in the next week;b. At
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
The probability of being dealt a full house in a hand of poker is approximately .0014. Find an approximation for the probability that in 1000 hands of poker, you will be dealt at least 2 full houses.
A total of 2n people, consisting of married couples, are randomly seated (all possible orderings being equally likely) at a round table. Let Ci denote the event that the members of couple are seated
In response to an attack of 10 missiles, 500 antiballistic missiles are launched. The missile targets of the antiballistic missiles are independent, and each antiballstic missile is equally likely to
A fair coin is flipped 10 times. Find the probability that there is a string of consecutive heads bya. Using the formula derived in the text;b. Using the recursive equations derived in the text.c.
At time 0, a coin that comes up heads with probability p is flipped and falls to the ground. Suppose it lands on heads. At times chosen according to a Poisson process with rate λ the coin is picked
Consider a roulette wheel consisting of 38 numbers 1 through 36, 0, and double 0. If Smith always bets that the outcome will be one of the numbers 1 through 12, what is the probability thata. Smith
Two athletic teams play a series of games; the first team to win 4 games is declared the overall winner. Suppose that one of the teams is stronger than the other and wins each game with probability
A fair coin is continually flipped until heads appears for the 10th time. Let X denote the number of tails that occur. Compute the probability mass function of X.
An urn contains 4 white and 4 black balls. We randomly choose 4 balls. If 2 of them are white and 2 are black, we stop. If not, we replace the balls in the urn and again randomly select 4 balls. This
A purchaser of transistors buys them in lots of 20. It is his policy to randomly inspect 4 components from a lot and to accept the lot only if all 4 are non-defective. If each component in a lot is,
An urn contains 10 red, black, and green balls. One of the colors is chosen at random (meaning that the chosen color is equally likely to be any of the colors), and then balls are randomly chosen
Show that1. E[Y] = P{Y > y} dy - P{Y < - y} dy
Consider the functionCould f be a probability density function? If so, determine C. Repeat if were given by C(2x-x³) 0
Show that X if has density function f, then E[g(X)] = g(x)f(x) dx 8
Define a collection of events Ea, 0 < a < 1, having the property that P(Ea) = 1 for all but P(nE.) = 0. %3D a
The standard deviation of X denoted SD(X), is given byFind SD(aX + b) if has variance σ2. SD(X) = Var(X)
The lifetime in hours of an electronic tube is a random variable having a probability density function given byf(x) = xe-x x ≥ 0Compute the expected
Let be a random variable that takes on values between 0 and c. That is, P{0 ≤ X ≤ c} = 1. Show thatVar(X) ≤ c2/4
Let be a standard normal random variable z, and let be a differentiable function with derivative g'.a. Show that E[g'(Z)] = E(Zg(Z)];b. Show that E[Zn+1] = nE[Zn-1].c. Find E[Z4].
A bus travels between the two cities and which are 100 miles apart. If the bus has a breakdown, the distance from the breakdown to city has a uniform distribution over (0, 100). There is a bus
Find the distribution of R = A sin θ, where is a fixed constant and is uniformly distributed on (- π/2, π/2). Such a random variable R arises in the theory of ballistics. If a projectile is fired
Let be a lognormal random variable (see Example 7e for its definition) and let be a constant. Answer true or false to the following, and then give an explanation for your answer.a. cY is lognormal;b.
The mode of a continuous random variable having density is the value of for which attains its maximum. Compute the mode of in cases (a), (b), and (c) of Theoretical Exercises 5.13.Theoretical
If is an exponential random variable with parameter λ, and c > 0, show that cX is exponential with parameter λ/c.
Compute the hazard rate function of X when X is uniformly distributed over (0, a).
The salaries of physicians in a certain specialty are approximately normally distributed. If 25 percent of these physicians earn less than $180,000 and 25 percent earn more than $320,000,
Verify thatVar(X) = α/λ2.when is a gamma random variable with parameters α and λ.
Suppose that the height, in inches, of a 25-year-old man is a normal random variable with parameters μ = 71 and σ2 = 6.25. What percentage of 25-year-old men are taller than 6 feet, 2 inches? What
Show that a plot of log(log(1 - F(x))-1) against log will be a straight line with slope β when F(·) is a Weibull distribution function. Show also that approximately 63.2 percent of all observations
LetY = (X - v/α)βShow that if is a Weibull random variable with parameters v, α, and β, then is an exponential random variable with parameter λ = 1 and vice versa.
Two types of coins are produced at a factory: a fair coin and a biased one that comes up heads 55 percent of the time. We have one of these coins but do not know whether it is a fair coin or a biased
Let be a continuous distribution function. If U is uniformly distributed on (0, 1), find the distribution function of Y = F-1(U), where F-1 is the inverse function of F. (That is, y = F-1(x) if F(y)
Consider the beta distribution with parameters (a, b). Show thata. When a > 1 and b > 1, the density is unimodal (that is, it has a unique mode) with mode equal to (a - 1)/(a +b - 2);b. When a
Let X be a continuous random variable having cumulative distribution function F. Define the random variable Y by Y = F(X). Show that is uniformly distributed over (0, 1).
Let X have probability density fX. Find the probability density function of the random variable defined by Y = aX + b.
Find the probability density function of Y = eX when is normally distributed with parameters μ and σ2. The random variable is said to have a lognormal distribution (since log has a normal
The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter λ = 1/2. What isa. What is the probability that a repair time exceeds 2 hours?b. the
If U is uniformly distributed on (0, 1) find the distribution of Y = - log(U).
If X is an exponential random variable with parameter λ and c > 0, find the density function of cX. What kind of random variable is cX.
Suppose X, Y have a joint distribution function F(x, y). Show how to obtain the distribution functions Fx(x) = P{X ≤ x} and FY(y) = P{Y ≤ y}.
If X is an exponential random variable with parameter λ = 1, compute the probability density function of the random variable defined by Y = log X.
The lung cancer hazard rate λ(t) of a -year-old male smoker is such thatλ(t) = .027 + .00025(t - 40)2 t ≥ 40Assuming that a 40-year-old male smoker survives all other hazards,
If is uniformly distributed over (a, b) find a and b E[X] = 10, Var(X) = 48.
If X is uniformly distributed over (0, 1), find the density function of Y= eX.
Two fair dice are rolled. Find the joint probability mass function of and whena. X is the largest value obtained on any die and is the sum of the values;b. X is the value on the first die and is the
Suppose that and are integer valued random variables and have a joint distribution function F(i, j) = P(X ≤ i, Y ≤ j).a. Give an expression, in terms of the joint distribution function, for P(X =
The severity of a certain cancer is designated by one of the grades 1, 2, 3, 4 with being the least severe and the most severe. If X is the score of an initially diagnosed patient and the score of
Consider a sequence of independent Bernoulli trials, each of which is a success with probability p. X1 Let be the number of failures preceding the first success, and let X2 be the number of failures
The joint probability density function of X and Y is given by a. Find c.b. Find the marginal densities of X and Y.c. Find E[X]. f(x, y) = c(y? - x²)e-y -ysxS y, 0 < y< ∞
The joint probability density function of and is given bya. Verify that this is indeed a joint density function.b. Compute the density function of X.c. Find P{X > Y}.d. Finde. Find E[X].f. Find
The joint probability density function of X and Y is given byFind(a) P{X < Y} and(b) P{X < a}. f(x, y) = e-(*+y) 0Sx< 0,0 sy< ∞
Let X1, X2, X3, X4, X5 be independent continuous random variables having a common distribution function and density function f, and set I = P{X1 < X2 < X3 < X4 < X5}a. Show that does not
The number of people who enter a drugstore in a given hour is a Poisson random variable with parameter λ = 10. Compute the conditional probability that at most 3 men entered the drugstore, given
A man and a woman agree to meet at a certain location about 12:30 P.M. If the man arrives at a time uniformly distributed between 12:15 and 12:45, and if the woman independently arrives at a time
Suppose that and are independent geometric random variables with the same parameter p.a. Without any computations, what do you think is the value ofb. Verify your conjecture in part (a). P{X = i|X+Y
The joint density of X and Y is given byAre and independent? If, instead, f(x, y) were given bywould and be independent? xe-(x+y) f(x, y) = lo x > 0, y > 0 otherwise
Suppose that 106 people arrive at a service station at times that are independent random variables, each of which is uniformly distributed over (0, 106). Let denote the number that arrive in the
Compute the density of the range of a sample of size from a continuous distribution having density function f.
Monthly sales are independent normal random variables with mean 100 and standard deviation 5.a. Find the probability that exactly of the next months have sales greater than 100.b. Find the
Let X1 and X2 be independent normal random variables, each having mean and variance σ2. Which probability is largera. b. P(X1 > 15) or Р(X, + X, > 25);
Suppose X and are independent normal random variables with mean 10 and variance 4. Find such thatP(X + Y > x) = P(X > 15).
Let X1, ..., Xn be independent uniform (0, 1) random variables. Let R = X(n) - X(1) denote the range and M = [X(n) + X(1)]/2 the midrange of X1, ..., Xn. Compute the joint density function of R
Suppose that has a beta distribution with parameters (a, b), and that the conditional distribution of given that X = x is binomial with parameters (n + m, x). Show that the conditional density of x
The discrete integer valued random variables X, Y, Z are independent if for all i, j, kShow that if are independent then and are independent. That is, show that the preceding implies that P(X =i,Y =
Two dice are rolled. Let X and Y denote, respectively, the largest and smallest values obtained. Compute the conditional mass function of Y given X = i, for i = 1, 2, ,..., 6. Are and independent?
The joint probability mass function of X and Y is given bya. Compute the conditional mass function of X given Y = i, i =1, 2.b. Are X and Y independent?c. Compute P{XY ≤ 3}, P{X + Y > 2}, P{X/Y
If 3 trucks break down at points randomly distributed on a road of length L, find the probability that no 2 of the trucks are within a distance of each other when d ≤ L/2.
Consider a sample of size 5 from a uniform distribution over (0, 1). Compute the probability that the median is in the interval (1/4, (3/4).
Let X(1), X(2), ... , X(n) be the order statistics of a set of independent uniform (0, 1) random variables. Find the conditional distribution of X(n) given that X(1), = s1, X(2) = s2, ... , X(n-1) =
If X and Y are independent and identically distributed uniform random variables on (0, 1), compute the joint density ofa. U = X + Y, V = X/Y;b. U = X, V = X/Y;c. U = X+ Y, V = X/(X + Y).
A total of balls, numbered 1 through 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. Finda. The expected number of
A group of men and 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.
An urn has black balls. At each stage, a black ball is removed and a new ball that is black with probability and white with probability is put in its place. Find the expected number of stages needed
Consider independent trials, the ith of which results in a success with probability Pia. Compute the expected number of successes in the trials—call it μ.b. For a fixed value of μ what choice of
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
Let X1, X2, ... be a sequence of independent and identically distributed continuous random variables. Let N ≥ 2 be such thatX1 ≥ X2 ≥ ··· ≥ XN-1 < XNThat is, is the point at which the
Prove the Cauchy–Schwarz inequality, namely, (E[XY])2 ≤ E[X2] E[Y2]

Showing 1 - 100 of 291