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probability and stochastic modeling

Fundamentals Of Probability 2nd Edition Saeed Ghahramani - Solutions

7 The moment-generating function of a random variable X is given by 1 Mx(t) = 1 < 1. (1-1)2 Find the moments of X.
6 The moment-generating function of X is given by Find P(X > 0). Mx(t) = exp().
5 Let the moment-generating function of a random variable X be given by Mx(t)= (e/-e-1/2) ift #0 Find the distribution function of X. if t = 0.
4 For a random variable X, suppose that Mx(t) = exp(212 +1). Find E(X) and Var(X).
3 The moment-generating function of a random variable X is given by Mx(t)==' + Find the distribution function of X. 1 021 3 +
2 The moment-generating function of a random variable X is given by Mx(t) = (+) 10 Find Var(X) and P(X 8).
1 Yearly salaries paid to the salespeople employed by a certain company are normally distributed with mean $27,000 and standard deviation $4900. What is the probability that the average wage of a random sample of 10 employees of this company is at least $30,000?
9 Let (X1, X2,...] be a sequence of independent Poisson random vari- ables, each with parameter 1. By applying the central limit theorem to this sequence, prove that lim k! 112
8 Let (X1, X2,) be a sequence of independent standard normal random variables. Let S X+X++X. Find = lim P(S, n+2n). Hint: See Exercise 8, Section 7.4.
7 A fair coin is tossed successively. Using the central limit theorem, find an approximation for the probability of obtaining at least 25 heads before 50 tails?
6 An investor buys 1000 shares of the XYZ Corporation at $50.00 per share. Subsequently, the stock price varies by $0.125 (1/8) every day, but unfor- tunately it is just as likely to move down as up. What is the most likely value of his holdings after 60 days? Hint: First calculate the distribution
5 Suppose that, whenever invited to a party, the probability that a person attends with his or her guest is 1/3, attends alone is 1/3, and does not attend is 1/3. A company has invited all 300 of its employees and their guests to a Christmas party. What is the probability that at least 320 will
4 A physical quantity is measured 50 times, and the average of these mea- surements is taken as the result. If each measurement has a random error uniformly distributed over (-1, 1), what is the probability that our result differs from the actual value by less than 0.25?
3 Each time that Jim charges an item to his credit card, he rounds the amount to the nearest dollar in his records. If he has used his credit card 300 times in the last 12 months, what is the probability that his record differs from the total expenditure by, at most, 10 dollars?
2 Let X1, X2, X,, be independent and identically distributed random variables, and let S, X+X2++X. For large n, what is the approximate probability that S,, is between E(S)-s, and E(S)+s,?
1 What is the probability that the average of 150 random points from the interval (0, 1) is within 0.02 of the midpoint of the interval?
15 Slugger Bubble Gum Company markets its best-selling brand to young baseball fans by including pictures of current baseball stars in packages of its bubble gum. In the latest series, there are 20 players included, but there is no way of telling which player's picture is inside until the package
14 Let (X1, X2, X3, ...) be a sequence of independent and identically dis- tributed exponential random variables with parameter A. Let N be a geo- metric random variable with parameter p independent of (X1, X2, X3,...). Find the distribution function of X.
13 Bus A arrives at a station at a random time between 10:00 AM. and 10:30 A.M. tomorrow. Bus B arrives at the same station at a random time between 10:00 AM. and the arrival time of bus A. Find the expected value of the arrival time of bus B.
12 Let the joint probability density function of X and Y be given by f(x, y) = ye-(+) if x>0, y >0 to otherwise. (a) Show that E(X) does not exist. (b) Find E(XY).
11 In terms of the means, variances, and the covariance of the random vari ables X and Y, find a and for which E(Y-a-BX)2 is minimum This is the method of least squares; it fits the "best" line y = a + x to the distribution of Y.
10 Let the joint probability density function of X and Y be given by f(x, y) = {* if 0 < y < x < elsewhere. (a) Find the marginal probability density functions of X and Y. (b) Determine the correlation coefficient of X and Y.
9 A random point (X, Y) is selected from the rectangle [0, /2] x [0, 1]. What is the probability that it lies below the curve y sin x?
8 Two green and two blue dice are rolled. If X and Y are the numbers of 6's on the green and on the blue dice, respectively, calculate the correlation coefficient of X Y and X + Y.
7 Let X and Y be jointly distributed with p(X, Y) = 2/3, x = 1, Var(Y) 9. Find Var(3X-5Y +7).
6 Let the joint probability density function of X, Y, and Z be given by f(x, y, z) = { [8xyz if 0
5 Determine the expected number of tosses of a die required to obtain four consecutive 6's.
4 In a town there are n taxis. A woman takes one of these taxis every day at random and with replacement. On average, how long does it take before she can claim that she has been in every taxi in the town? Hint: The final answer is in terms of a,, = 1+1/2+...+1/n.
3 Let the joint probability density function of random variables X and Y be 3x3 + xy if 0 x1, 0 y 2 f(x, y) = 3 0 elsewhere. Find E(X + 2XY).
2 Let the probability function of a random variable X be given by 2x-2 if 1
1 A hospital nurse mixes the 10 pills of 10 patients accidentally. Suppose that she gives a pill at random to each patient from the mixed-up batch. If none of the 10 pills are of the same type, what is the expected number of patients who will get their prescribed pills?
7 Let the joint probability density function of random variables X and Y be bivariate normal. Show that if oxy, then X + Y and X - Y are independent random variables. Hint: Show that the joint probability density function of X+X and X-Y is bivariate normal with correlation coefficient 0.
6 Let Z and W be independent standard normal random variables. Let X and Y be defined by Y = 0 [pZ+ 1-p W] + H2 where , > 0, < 141 142
5 Let the joint probability density function of two random variables X and Y be given by
4 Let f(x, y) be a joint bivariate normal probability density function. De- termine the point at which the maximum value of f is obtained.
3 Let the joint probability density function of X and Y be bivariate normal. For what values of a is the variance of aX + Y minimum?
2 The joint probability densitty function of X and Y is bivariate normal with ox=y=9, x=My = 0, and p = 0. Find (a) P(X 6, Y12); (b) P(X2 Y2
28 (The Wallet Paradox) Consider the following "paradox" given by Martin Gardner in his book "Aha! Gotcha," W. H. Freeman and Company, New York, 1981.1 Let X be the height of a man and Y the height of his daughter (both in inches). Suppose that the joint probability density function of X and Y is
27 Let the joint probability density function of random variables X and Y be given by 1 if lyx, 0 < x
26 There are prizes in 10% of the boxes of a certain type of cereal. Let X be the number of boxes of such cereal that Kim should buy to find a prize. Let Y be the number of additional boxes of such cereal that she should purchase to find another prize. Calculate the joint probability function of X
25 A bar of length is broken into three pieces at two random spots. What is the probability that the length of at least one piece is less than /20?
24 Alvic, a marksman, fires seven independent shots at a target. Suppose that the probabilities that he hits the bull's-eye, he hits the target but not the bull's-eye, and he misses the target are 0.4, 0.35, and 0.25, respectively. What is the probability that he hits the bull's-eye three times,
23 A fair die is tossed 18 times. What is the probability that each face appears three times?
22 If F is the probability distribution function of a random variable X, is G(x, y) = F(x)+ F(y) a joint probability distribution function?
21 The joint probability density function of random variables X, Y, and Z is given by f(x, y, z) = {(x + y +22) if 0 x, y, z 1 otherwise. (a) Determine the value ofc. (b) Find P(X
20 Let X and Y be two independent uniformly distributed random variables over the intervals (0, 1) and (0, 2), respectively. Find the probability density function of X/Y.
19 Suppose that n points are selected at random and independently inside the cube =((x, y, z): a xa, -a ya, -a za). Find the probability that the distance of the nearest point to the center is at least r (r
18 A point is selected at random from the bounded region between the curves y = x-1 and y = 1-x. Let X be the the x-coordinate, and let Y be the y-coordinate of the point selected. Determine if X and Y are independent.
17 Let the joint probability density function of X and Y be given by [cx(1-x) if0x y 1 f(x, y) = {ex(1-x) (a) Determine the value ofc. otherwise. (b) Determine if X and Y are independent.
16 Let X be the smallest number obtained in rolling a balanced dien times. Calculate the probability distribution function and the probability function of X.
15 From an ordinary deck of 52 cards, cards are drawn successively and with replacement. Let X and Y denote the number of spades in the first 10 cards and in the second 15 cards, respectively. Calculate the joint probability function of X and Y.
14 Let X and Y be continuous random variables with the joint probability density function f(x, y) = {0 if y> 0, 0 < x
13 For =((x, y): 0 < x+y
12 Let the joint probability distribution function of the lifetimes of two brands of lightbulbs be given by F(x, y) = {(1-e) (1-e) if x>0, y>0 otherwise. Find the probability that one lightbulb lasts more than twice as long as the other.
11 A fair coin is flipped 20 times. If the total number of heads is 12, what is the expected number of heads in the first 10 flips?
10 Three concentric circles of radii r. 2, and 3, r>r2> 13, are the boundaries of the regions that form a circular target. If a person fires a shot at random at the target, what is the probability that it lands in the middle region?
9 Prove that the following cannot be the joint probability distribution func- tion of two random variables X and Y. 1 if x + y 1 F(x, y) = {0 if x + y < 1.
8 Let the joint probability density function of X and Y be given by xy+y if 0
7 Let the joint probability density function of X and Y be given by Jc/x if0
6 An urn contains 100 chips of which 20 are blue, 30 are red, and 50 are green. Suppose that 20 chips are drawn at random and without replacement. Let B, R, and G be the number of blue, red, and green chips, respectively. Calculate the joint probability function of B, R, and G.
5 Calculate the probability function of the number of spades in a random bridge hand that includes exactly four hearts and three clubs.
4 Suppose that three cards are drawn at random from an ordinary deck of 52 cards. If X and Y are the numbers of diamonds and clubs, respectively, calculate the joint probability function of X and Y..
3 Calculate the probability function of the number of spades in a random bridge hand that includes exactly four hearts.
2 A fair die is tossed twice. The sum of the outcomes is denoted by X and the largest value by Y. Calculate the joint probability function of X and Y and the marginal probability functions of X and Y.
1 The joint probability function of X and Y is given by the following table. x y 1 2 3 1246 0.05 0.25 0.15 0.14 0.10 0.17 0.10 0.02 0.02 Find P(XY 6).
12 Let X1, X2, and X3 be independent random variables from (0, 1). Find the probability density function and the expected value of the midrange of these random variables [X(1) + X(3)]/2.
11 Let X1, X2, X, be n independently randomly selected points from the interval (0, 0), 0>0. Prove that n-1 E(R)= -0. n+1 where R = X(n) - X() is the range of these points. Hint: Use part (a) of Exercise 10. Also compare this with Exercise 18, Section 8.4.
10 Let X1, X2, X, be a random sample of size n from a population with continuous probability distribution function F and probability density function f.(a) Calculate the probability density function of the sample range, R = X(n) - X(1) (b) Use (a) to find the probability density function of the
9 Let X, and X2 be two independent exponential random variables each with parameter. Show that X() and X(2) - X(1) are independent.
8 Let X and Y be independent (strictly positive) exponential random vari- ables each with parameter A. Are the random variables X + Y and X/Y independent? Exercises 9 through 12 concern the order statistics of sets of ran- dom variables and refer to Section 8.5.Exercises 9 through 12 concern the
7 Let X and Y be independent (strictly positive) gamma random variables with parameters (1, 2) and (r2, 2), respectively. Define U=X+Y and V = x/(X + Y). (a) Find the joint probability density function of U and V. (b) Prove that U and V are independent. (c) Show that U is gamma and V is beta.
6 Prove that if X and Y are independent standard normal random variables, then X + Y and X Y are independent random variables. This is a special case of the following important theorem. Let X and Y be independent random variables with a common distribution F. The random variables X + Y and X - Y
5 Let X and Y be independent random variables with common probability density function if x>0 elsewhere. Find the joint probability density function of UX + Y and V = ex.
4 Let X and Y be independent random variables with common probability density function f(x) = {1/x 1/x if x 1 elsewhere. Calculate the joint probability density function of UX/Y and V XY.
3 Let X N(0, 1) and Y N(0, 1) be independent random variables. Find the joint probability density function of R = X2+ Y2 and 0 = arctan(Y/X). Show that R and are independent. Note that (R, 0) is the polar coordinate representation of (X, Y).
2 Let X and Y be two positive independent continuous random variables with the probability density functions f(x) and f(y), respectively. Find the probability density function of U = X/Y. Hint: Let VX; find the joint probability density function of U and V. Then calculate the marginal probability
1 Let X and Y be independent random numbers from the interval (0, 1). Find the joint probability density function of U-2 In X and V = -2 In Y.
12 Let X be a continuous random variable with distribution function F and density functionf. Find the distribution function and the density function of Y = |X|.
11 The lifetime (in hours) of a lightbulb manufactured by a certain company is a random variable with probability density function 0 f(x) = 5 x 10 if x 500 if x > 500. Suppose that, for all nonnegative real numbers a andb, the event that any lightbulb lasts at least a hours is independent of the
10 Let X be a continuous random variable with set of possible values (x: 0 < xa) (where a
9 Prove or disprove: If a 1, ; 0, Vi, and (fi) is a sequence of density functions, then a fi is a probability density function.
8 Let F, the distribution of a random variable X be defined by 0 x
7 The probability density function of a continuous random variable X is (30x2(1-x) if 0 < x
6 Let X be a random variable with density function 4x/15 1x2 f(x) = {4x/15 0 otherwise. Find the density functions of Y = ex, Z = x, and W = (X-1).
5 Does there exist a constant c for which the following is a density function? c/(1+x) ifx>0 = { c/ (1 + x) f(x) = otherwise.
4 Let X be a random variable with density function f(x)= Find P(-2X < 1). 2
3 Let X be a continuous random variable with density function f(x) = 6x(1-x), 0 < x
2 Let X be a continuous random variable with the probability density func- tion f(x) = {2/x [2/x if x > 1 otherwise. 0 Find E(X) and Var(X) if they exist.
1 Let X be a random number from (0, 1). Find the probability density function of Y=1/X.
19 Let X be a continuous random variable with probability density functionf. Show that if E(X) exists; that is, if x f(x) dx < o, then lim x P(X x) = lim xP(X>x) = 0. x-x
18 Let X be the random variable introduced in Exercise 12. Applying the results of Exercise 16, calculate Var(X).
17 Let X be a continuous random variable. Prove that P(|X|n) E(XI) 1+ P(|X|n). These important inequalities show that E(X) < if and only if the series P(Xn) converges. Hint: By Exercise 16, *** E(X) = P(X)>t) = P(\X\ > t) dt = * * * P (|X|>t)dt. Note that on the interval [n, n+1), P(Xn+1)P(|X|>t)
16 Let X be a nonnegative random variable with distribution function F. Define 1(t)= = { b 1 if X > t 10 otherwise.(a) Prove that I(t)dt = X. (b) By calculating the expected value of both sides of part (a), prove that E(X) = (1-F(1)]dt. This is a special case of Theorem 6.2. (c) For r> 0, use part
15 Let X be a continuous random variable with density functionf. A number t is said to be the median of X if P(X 1) = P(X 1) = 1
14 Let X be a continuous random variable with the probability density func- tion f(x) = {0 (1/7)x sinx if 0 < x < otherwise. Prove that +1 E(X+1)+(n+1)(n+2)E(X") ="+.
13 For n 1, let X,, be a continuous random variable with the probability density function f(x) = = {C/x+ C/x+ if x C otherwise. X's are called Pareto random variables and are used to study income distributions. (a) Calculatec. n 1. (b) Find E(X,), n 1. (c) Determine the density function of Z = In
12 Suppose that X, the interarrival time between two customers entering a certain post office, satisfies P(X1)=ae + Be 10, where a + 8 = 1, a 0, 80, > 0, >0. Calculate the expected value of X. Hint: For a fast calculation, use Remark 3 following Theorem 6.2.
11 Let X be a random variable with the probability density function f(x) 1 (1+x)' Prove that E(X) converges if 0 < a
10 Let X be a random variable with probability density function f(x) = -x Calculate Var(X). B
9 A right triangle has a hypotenuse of length 9. If the probability density function of one side's length is given by f(x)= [x/6 if 2

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