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Introduction To Probability And Statistics For Science Engineering And Finance 1st Edition Walter A. Rosenkrantz - Solutions

Problem 5.62 Let (X, Y ) be uniformly distributed over the unit circle {(x, y) : x2 + y2 ≤ 1}. Its joint distribution function is given by f(x, y) = 1 π , for x2 + y2 ≤ 1 f(x, y)=0, elsewhere. Compute: (a) P(X2 + Y 2 ≤ 1/4). (b) P(X>Y ). (c) P(X = Y ). (d) P(Y < 2X). (e) Let R = X2 + Y 2.
Problem 5.61 Let the random vector (X, Y ) have the joint density function f(x, y) = xe−xy−x, for x > 0,y> 0 f(x, y)=0, elsewhere. Compute: (a) fX(x), µX, σX . (b) fY (y), µY , σY . (c) Are X and Y independent?
Problem 5.60 Given f(x, y) = c(x + 2y), 0
Problem 5.59 Let (X, Y ) have the joint density given by f(x, y)=1/4, −1 2X) (f) P(Y = X) (g) P(X + Y < 0.5) (h) P(X + Y < 1) (i) P(X + Y < 1.5)
Problem 5.58 High priority freight service (Source: Wayne Nelson (1982), Applied Life Data Analysis, John Wiley and Sons, Inc., New York). A high priority freight train required three locomotives for a one–day run. If any one of these locomotives failed the train was delayed and the railroad had
Problem 5.57 For the system displayed below compute: (a) the system function X in terms of X1,...,Xn and (b) the reliability of the system, assuming P(Xi = 1) = p and the components are indepen[1]dent 2 3
Problem 5.56 For the system displayed below compute: (a) the system function X in terms of X1,...,Xn and (b) the reliability of the system, assuming P(Xi = 1) = p and the components are indepen[1]dent. 3 2
Problem 5.55 By considering the various possibilities verify that the 2-out-of-3 system function is given by Equation 5.127.
Problem 5.54 Refer to Example 5.32. Derive the formula for Cov(v(t), v(s)) given in Equation 5.118.
Problem 5.53 Let X(t) be a Poissont process with intensity λ. (a) Show that limt→∞ E X(t) t − λ 2 = 0. (b) Use Chebyshev’s inequality and part (a) to show that for every d > 0 limt→∞ P X(t) t − λ > d = 0.
Problem 5.52 A critical component of a system needs to function continuously without failure for 5 days. The distribution of the component’s lifetime is exponential with expected lifetime E(T )=2.5. As soon as he component fails it is immediately replaced by a spare component with the same
Problem 5.51 Let Wn have an Erlang distribution as described in Equation 5.120. Verify the formula for the pdf displayed in Equation 5.121.
Problem 5.50 An inventory control model. Consider a warehouse containing an inventory of 10 high voltage transformers. If at the end of the week the inventory level falls below 2 (≤ 2), an order is placed to return the inventory level to 10. Suppose orders per week arrive according to a Poisson
Problem 5.49 X(t) is a Poisson process with intensity λ = 1.5. Compute: (a) P(X(2) ≤ 3). (b) P(X(2) = 2, X(4) = 6). Hint: X(2) and X(4)−X(2) are independent random variables.
Problem 5.48 Telephone calls arrive according to a Poisson process X(t) with intensity λ = 2 per minute. Compute: (a) E(X(1)), E(X(5)). (b) E(X(1)2), E(X(5)2). (c) What is the probability that no phone calls arrive during the interval 9:10 to 9:12?
Problem 5.47 Refer to Example 5.28. Suppose the current price of a stock is $40, its volatility σ = 0.30, and the current interest rate is 5.5%, compounded continuously. What is the price of a put option with strike price $36 and an expiration date of (a) three months? (b) six months? Sections
Problem 5.46 The current price of a stock is $94, and the price of one three-month Eu[1]ropean call option with a strike price K = 95 is $4.70(per share). You are offered two investment strategies: • Strategy 1: Buy 100 shares @ $94 per share: total investment = $9400, or • Strategy 2: Buy 2000
Problem 5.45 A stock price is currently $100. Over each of the next two six-month periods it is expected to go up by 10% or down by 10%. The risk free interest rate is 8% per year. (a) What is the value of a one-year European call option with a strike price of $100? (b) What is the value of a
Problem 5.44 The price S(t) (t measured in years) of a stock is governed by a geometric Brownian motion with parameters µ = 0.09, volatility σ = 0.18, and initial price S0 = $50. (a) An option on the stock has a maturity date of 3 months. Compute the expected price of the stock on the maturity
Problem 5.43 For the situation considered in Problem 5.42: (a) Find suitable parameters of the binomial lattice model that approximates the stock price (not the option price) with a basic time period of 3 months. (b) Draw the binomial tree for the binomial lattice model of part (a), (similar to
Problem 5.42 The price S(t) (t measured in years) of a stock is governed a geometric Brownian motion with expected return µ = 0.16, volatility σ = 0.35, and current price S0 = $38. (a) An option on the stock has a maturity date of 6 months. Compute the expected price of the stock on the maturity
Problem 5.41 Dealer A offers 0% financing for a $10, 000 car. The customer pays $1, 000 down and $300 per month for the next 30 months. Dealer B does not give 0% financing but takes $1, 000 off the price. Which deal is better (for the customer) if the annual interest rate r is: (a) r = 10%?Hint:
Problem 5.40 This is a continuation of Problem 1.10. An important performance measure of a stock market portfolio is to compute its annualized returns for the best three year period, as well as the worst three year period, over the past ten years. For the S&P500 the best three year period (over the
Problem 5.39 An International Equity Fund reported the annual returns for the years 2001-2005 in the Table 5.9.year 2001 2002 2003 2004 2005 returns -9.04% -8.25% 26.13% 22.94% 11.55%(a) At the beginning of 2001 Ms. Allen invests $10,000 in this mutual fund where it remains for the next five years.
Problem 5.38 A lottery pays the winner $10 million. The prize money is paid out at the annual rate of $500,000 each year, for twenty years, with the first payment at the beginning of the first year. What is the present value of this prize at: (a) 4% interest? (b) 8% interest?
Problem 5.37 The estimated standard deviations and correlations for three stocks are given in the table below. Stock σ A B C A 0.30 1 −0.63 0.52 B 0.20 −0.63 1 −0.60 C 0.36 0.52 −0.60 1 (a) Compute the standard deviation of a portfolio composed of 50% of B and 50% of C. (b) Compute the
Problem 5.36 Given the correlation between securities A and B is ρ(A, B)=0.20, with expected returns and standard deviations listed below, compute the expected return and stan[1]dard deviation for each of the following portfolios. (a) Portfolio 1: 50% of stock A and 50% of stock B. (b) Portfolio
Problem 5.35 Given the correlation between securities A and B is ρ(A, B)=0.30, with expected returns and standard deviations listed below, compute the expected return and stan[1]dard deviation of a portfolio composed of 40% of A and 60% of B. Security Return σ Proportion A 0.10 0.20 0.40 B 0.15
Problem 5.34 Refer to the resort island stock market (Example 5.21) with the modified joint distribution function given below. (a) Compute the expected returns, standard deviations, of RA, RB and the correlation be[1]tween them. (b) Compute the expected return RC and standard deviation of a
Problem 5.33 Verify that choosing the parameters of the binomial lattice model as in Equation 5.60 yields Equations 5.62 and 5.63 for the mean and variance.
Problem 5.32 Suppose the distribution of a stock price S(t) (t measured in years) is gov[1]erned by a geometric Brownian motion with parameters µ = 0.20, volatility σ = 0.30, and initial price S0 = $60. Compute: (a) E(stock price after 6 months). (b) σ(stock price after 6 months). (c) Find the
Problem 5.31 Suppose the distribution of a stock price S(t) (t measured in years) is gov[1]erned by a geometric Brownian motion with parameters µ = 0.15, volatility σ = 0.25, and initial price S0 = $40. Compute: (a) E(stock price after 1 year). (b) σ(stock price after 1 year). (c) Find the
Problem 5.30 The price S(t) (t measured in years) of a stock is governed by a geometric Brownian motion with parameters µ = 0.12, volatility σ = 0.20, and initial price S0 = $75. Compute: (a) E(stock price after 6 months). (b) σ(stock price after 6 months). (c) Show that the probability of a
Problem 5.29 Suppose X1, X2 are independent Poisson random variables with parameters λ1, λ2. Show that X1 + X2 is a Poisson random variable with parameter λ = λ1 +λ2. Hint: Compute the moment generating function MX1+X2 (s) using Theorem 5.2 and Equation 3.69.
Problem 5.28 A mathematical model for a random sine wave X(t) assumes that X(t) = A cos(t + Θ) where A (the amplitude) and Θ (the phase) are mutually independent random variables, Θ is uniformly distributed over [0, 2π], and E(A) = µ1 and E(A2) = µ2 are both finite. (a) Show that E(X(t)) = 0
Problem 5.27 Let X denote the mean of a random sample of size 16 taken from an expo[1]nential distribution with parameter θ = 1/4. (a) Compute the mean and variance of X. (b) Use the central limit theorem to compute approximate values for P(2.4 < X < 5.6).
Problem 5.26 Suppose the sequence X1,...,X12 is a random sample taken from the uni[1]form distribution on the interval [0, 1](Equation 4.11). (a) Compute the mean and variance of the sample total T = X1 + ... + X12. (b) What is the largest value that T can assume? The smallest? (c) Use the central
Problem 5.25 Derive the properties 5.37 and 5.38 of the correlation coefficient.
Problem 5.24 Let X, Y,W be random variables with finite means and variances. (a) Show that Cov(aX, cY ) = acCov(X, Y ). (b) Show that Cov(X + Y,W) = Cov(X,W) + Cov(Y,W). (c) Give the details of the derivation of Equation 5.34.
Problem 5.23 The random variables X1,...,X40 are independent with the same probabil[1]ity function given by x −1 0 1 f(x) 0.2 0.2 0.6 (a) Compute the mean and variance of the sample total T = X1 + ... + X40. (b) What is the largest value that T can assume? The smallest? (c) What does the central
Problem 5.22 Suppose X and Y are random variables with E(X)=9 , E(Y )=6; E(X2) = 405 , E(Y 2) = 232 , E(XY ) = −126.Using only this information, compute: (a) V (X) and V (Y ) (b) Cov(X, Y ) and ρ(X, Y ) (c) V (X + Y ) (d) Are X and Y independent?
Problem 5.21 The random variables X and Y are independent with E(X)=2, V (X) = 9, E(Y ) = −3, V (Y ) = 16. Compute: (a) E(3X − 2Y ) (b) V (3X) (c) V (−2Y ) (d) V (3X − 2Y )
Problem 5.20 Suppose that math SATs are N(500, 1002) distributed. Six students are selected at random. What is the probability that of their math SAT scores: 2 are less than or equal to 450, 2 are between 450 and 600, and 2 are greater than or equal to 600?
Problem 5.19 The “placebo effect” refers to treatment where a substantial proportion of the patients report a significant improvement even though the treatment consists of nothing more than giving the patient a sugar pill or some other harmless inert substance. Ten patients are given a new
Problem 5.18 The output of steel plate manufacturing plant is classified into one of three categories: no defects, minor defects, and major defects. Suppose that P(no defects)=0.75, P(minor defects)=0.20, and P(major defects)=0.05. A random sample of 20 steel plates is inspected. Let Y1, Y2, Y3
Problem 5.17 A die is thrown 6 times. Let Yi = the number of times that i appears. (a) Compute P(Y1 = 1,...,Y6 = 1). (b) Compute P(Y1 = 2, Y3 = 2, Y5 = 2). (c) Compute E(Yi), i = 1,..., 6.
Problem 5.16 Let (X1,...,Xn) be mutually independent random variables having the dis[1]crete uniform distribution (see Equation 3.16). (a) Compute the probability function for the random variable Mn defined by Mn = max(X1,...,Xn). Hint: Compute P(Mn ≤ x). (b) Compute the probability function for
Problem 5.15 Compute the probability function of X + Y , where X and Y are mutually independent random variables with probability functions x 0 1 2 f(x) 0.25 0.5 0.25 y 0 1 g(y) 0.5 0.5
Problem 5.14 A fair die is thrown n times and the successive outcomes are denoted by X1,...,Xn. Let Mn = max(X1,...,Xn). Compute the conditional probabilities pij defined by pij = P(Mn = j|Mn−1 = i), (i = 1,..., 6; j = 1,..., 6). Display your answers in the format of a 6 × 6 matrix, with first
Problem 5.13 Weird dice: Consider 2 six-sided dice whose faces are labeled as follows: Die 1 = {1, 2, 2, 3, 3, 4}, Die 2 = {1, 3, 4, 5, 6, 8}. Let Yi denote the number that turns up when the ith die is rolled. Assume that the two random variables Y1, Y2 are independent. Compute the probability
Problem 5.12 Non-transitive dice: Consider 3 six-sided dice whose faces are labeled as follows: Die 1 = {5, 7, 8, 9, 10, 18}; Die 2 = {2, 3, 4, 15, 16, 17}; Die 3 = {1, 6, 11, 12, 13, 14}. Let Yi denote the number that turns up when the ith die is rolled. Assume that the three random variables Y1,
Problem 5.11 A bridge hand is a set of 13 cards selected at random from a deck of 52. Compute the probability of: (a) getting x hearts; (b) getting y clubs; (c) getting x hearts and y clubs.
Problem 5.10 Consider the experiment of drawing in succession and without replacement two coins from an urn containing two nickles and three dimes. Let X1 and X2 denote the monetary values (in cents) of the first and second coins drawn respectively. Compute: (a) The joint pmf of the random
Problem 5.9 Refer to Example 5.5. (a) Compute the joint and marginal probability functions of (Y, Z). (b) The monetary value (in cents) of the coins in the sample is given by W = 5X+10Y +25Z. Compute E(W).
Problem 5.8 Two numbers are selected at random and without replacement from the set {1, 1, 1, 1, 0, 0, 0, 0, 0, 0}. Let Xi, (i = 1, 2) denote the ith number that is drawn. (a) Compute the joint pmf of (X1, X2). Are (X1, X2) mutually independent? (b) Compute the pmf of X1 + X2.
Problem 5.7 Let (X, Y ) have the joint pmf given in Problem 5.6. Compute: (a) E(X), E(X2), V (X). (b) E(Y ), E(Y 2), V (Y ). (c) E(XY ). (d) Cov(X, Y ) and ρ(X, Y ).
Problem 5.6 The (discrete) joint pmf of X, Y is f(x, y) = cxy, x = 1, 2, 3; y = 1, 2. (a) Show that c = 1/18. (b) Display the joint pmf and marginal pmfs in the format of Table 5.2. (c) Compute P(Y = 1|X = x), x = 1, 2, 3. (d) Are X, Y independent random variables? Justify your answer. (e) Using
Problem 5.5 Let (X, Y ) have the joint pmf given in Problem 5.4. Compute: (a) E(X), E(X2), V (X). (b) E(Y ), E(Y 2), V (Y ). (c) E(XY ). (d) Cov(X, Y ) and ρ(X, Y ).
Problem 5.4 The (discrete) joint pmf of X, Y is f(x, y) = c(2x+y), x = 1, 2, 3; y = 1, 2, 3. (a) Show that c = 1/54. (b) Display the joint pmf and marginal pmfs in the format of Table 5.2. (c) Using the joint pmf computed in part (b) compute the pmf of X + Y. (d) Compute P(X ≤ Y ). (e) Compute
Problem 5.3 Let X and Y have the joint pmf given in Table 5.4. Compute: (a) P(Y ≤ 1|X = x) for x = 0, 1, 2. (b) P(Y = y|X = 0) for y = 0, 1, 2.
Problem 5.2 Let (X, Y ) have the joint pmf displayed in Problem 5.1. Compute the fol[1]lowing probabilities: (a) P(X ≤ 1, Y ≤ 3). (b) P(X > 1,Y > 3).(c) P(X = 2,Y > 2). (d) P(Y = y|X = 0) : y = 2, 3, 4, 5. (e) P(Y = y|X = 1) : y = 2, 3, 4, 5. (f) P(Y = y|X = 2) : y = 2, 3, 4, 5.
Problem 5.1 Let (X, Y ) have the joint pmf specified in the table below: X\ Y 2 3 4 5 0 1/24 3/24 1/24 1/24 1 1/12 1/12 3/12 1/12 2 1/12 1/24 1/12 1/24 (a) Compute the marginal probability functions. (b) Are X and Y independent? (c) Compute µX and σX. (d) Compute µY and σY .
Problem 4.83 Which of the following statements are true. If false, give a counter exam[1]ple. (a) E(X2)=(E(X))2 (b) E(1/X)=1/E(X) (c) If E(X)=0, then X = 0.
Problem 4.82 Is the following statement true or false? Suppose the random variables X and Y have the same distribution function F(x) = P(X ≤ x) = P(Y ≤ x). Then X = Y . If true, give a proof; if false, produce a counter example.
Problem 4.81 The first two moments and variance of the exponential distribution with parameter θ were derived in Example 4.5. These formulas can also be derived using the moment generating function technique. In detail: (a) Using Proposition 4.2 compute M X(s) and verify that M X(0) = 1 θ . (b)
Problem 4.80 The random variable X has pdf F given by f(x) = 2(1 − x), 0
Problem 4.79 The random variable X has pdf f given by f(x) = (1 + x), −1 ≤ x ≤ 0, f(x) = (1 − x), 0 ≤ x ≤ 1, f(x)=0, elsewhere. Compute the df and pdf of Y = X2.
Problem 4.78 The random variable T has df F given by F(t)=0, t ≤ 1, F(t)=1 − 1 t2 , t ≥ 1. Compute the df and pdf of X = T −1.
Problem 4.77 Let X = tanU, where U is uniformly distributed on the interval [−π/2, π/2]. (a) Show that FX(x) = 1 2 + 1 π arctan x. (b)Show that fX(x) = 1 π(1 + x2 .
Problem 4.76 Let Y have pdf g(y) = (1 + y)−2,y > 0 and g(y)=0, elsewhere. (a) Compute the df G(y) for all y. (b) Let X = 1/Y . Verify that X and Y have the same df.
Problem 4.75 Let R = √ X where X has an exponential distribution with parameter θ = 1/2. Show that R has a Rayleigh distribution: FR(r) = r 0 te−t 2/2 dt, r > 0, FR(r)=0, elsewhere.
Problem 4.74 The random variable U is uniformly distributed on [0, 1]. Show that Y = −θ−1 lnU has an exponential distribution with parameter θ, i.e., P(Y
Problem 4.73 Let U be a uniformly distributed random variable on the interval [−1, 1]. Let Y = 4 − U2. (a) Compute G(y) = P(Y ≤ y) for all y. (b) Compute the pdf g(y) for all y.
Problem 4.72 Let U be a uniformly distributed random variable on the interval [−1, 1]. Let X = U2. (a) Compute df F(x) = P(X ≤ x) for all x. (b) Compute the pdf f(x) for all x.
Problem 4.71 Suppose the distribution of the random variable X is Beta(α, β). Show that the distribution of 1 − X is Beta(β,α).
Problem 4.69 Show that Z (the standard normal random variable) and −Z have the same distribution. Problem 4.70 Suppose the random variables X and −X have the same distribution. Show that E(X)=0.
Problem 4.68 Show that the function O(p) defined by O(p) = P(X ≤c) = c x=0 b(x; n, p) is a decreasing function of p.
Problem 4.67 Derive the formulas for the mean and variance of the beta distribution displayed in Equation 4.70. Hint: E(X) = B(α + 1, β)/B(α, β) and E(X2) = B(α + 2, β)/B(α, β).
Problem 4.66 Suppose the proportion X of arsenic in copper has a beta distribution with α = 1.01, β = 100. Compute E(X) and V (X).
Problem 4.65 Graph the beta distribution for: (a) α = 2, β = 1. (b) α = 1, β = 2. (c) α = 1, β = 1. (d) α = 0.5, β = 1.
Problem 4.64 Suppose the proportion X of correct answers a student gets on a test has a beta distribution with parameters α = 4, β = 2. (a) Compute the cdf FX (x). (b) If a passing score is is 50%, what is the probability that a student passes? (c) If 100 students take the test, what is the
Problem 4.63 Consider the beta pdf f(x) = Cx2(1 − x)3, 0 ≤ x ≤ 1; f(x)=0, elsewhere. Identify the parameters α, β, and find the constant C. (Refer to Equations 4.68 and 4.69.)
Problem 4.62 Suppose that X has a beta distribution with parameters α = 3, β = 2. Compute E(X) and V (X) using Equation 4.70.
Problem 4.61 Using Equations 4.68 and 4.58 compute: (a) B(1/2, 1). (b) B(3, 2). (c) B(3/2, 2).
Problem 4.60 The lifetime T , measured in hours, of a disk drive has a Weibull distribution with parameters α = 0.25, β = 1000. (a) Compute E(T ) and V (T ). (b) What is the probability that the lifetime exceeds 6000 hours? (c) Compute the probability of the disk drive failing on a 10, 000 hour
Problem 4.59 An engine fan life, measured in hours, has a Weibull distribution with α = 1.053, β = 26, 710. Compute: (a) the median life. (b) the proportion that fail on an 8, 000 hour warranty.
Problem 4.58 The lifetime, measured in years, of a TV picture tube has a Weibull distri[1]bution with parameters α = 2, β = 3. (a) Compute the proportion of tubes that will fail on a: one year, two year, and three year warranty. (b) Compute the expected profit per TV tube if the warranty is for
Problem 4.57 The lifetime T of a tire, measured in units of 1, 000 miles, has a Weibull distribution with parameters α = 0.5, β = 15. Compute: (a) the median mileage of the tire. (b) the expected mileage E(T ). (c) P(T > 10).
Problem 4.56(a) If X has a gamma distribution with E(X) = µ1 and E(X2) = µ2 show that α = µ2 1 µ2 − µ2 1 and β = µ2 − µ2 1 µ1 . (b) Use the result of part (a) to find the parameters of the gamma distribution when µ1 = 2, µ2 = 5.
Problem 4.55 Derive the formulas for the mean and variance of the gamma distribution given in Equation 4.61.
Problem 4.54 Derive Equation 4.58 by performing a suitable integration by parts.
Problem 4.53(a) Show that Γ(1/2) = √π by making the change of variables x = u2/2 in the integral Γ(1/2) = ∞ 0 x−1/2e−x dx. (b) Using the result of part (a) compute Γ(3/2), Γ(5/2).
Problem 4.52 Consider the gamma pdf f(x) = Cx1/2e−x/2,x> 0; f(x)=0, elsewhere. Identify the parameters α, β, and find the constant C. (Refer to Equation 4.55.)
Problem 4.51 Suppose the probability that a person cancels his hotel reservation is 0.1 and the hotel’s capacity is 1, 000 rooms. In order to minimize the number of empty rooms the hotel routinely overbooks. How many reservations above 1, 000 should it accept if it wants the probability of
Problem 4.50 Suppose that the velocity V of a particle of mass m is N(0, 1) distributed; denote its kinetic energy by K = mv2/2. Show that E(K) = m/2 and V (K) = m2/2.
Problem 4.49 The number of chocolate chips per cookie is assumed to have a Poisson distribution with λ = 4. From a batch of 500 cookies, let Y denote the number of cookies without chocolate chips. (a) Describe the exact distribution of Y ; name it and identify the parameters. (b) Find E(Y ) and V
Problem 4.48 The natural recovery rate for a certain disease is 40%. A new drug is developed and the manufacturer claims that it boosts the recovery rate to 80%. Two hundred patients are given the drug. What is the probability that at least 100 patients recover if: (a) the drug is worthless? (b) it
Problem 4.47 A certain type of seed has a probability of 0.7 of germinating. In a package of 100 seeds let Y denote the number of seeds that germinate. (a) Give the formula for the exact distribution of Y . (b) Calculate the probability that at least 75% germinate. (c) Calculate the probability
Problem 4.46 Let Y denote the number of times a 1 appears in 720 throws of a die. Using the normal approximation to the binomial approximate the following probabilities: (a) P(Y ≥ 130) (b) P(Y ≥ 140) (c) P(|Y − 120| ≥ 20) (d) P(105
Problem 4.45 Let X denote the number of heads that appear when a fair coin is tossed 300 times. Use the normal approximation to the binomial to estimate the following probabilities: (a) P(X ≥ 160) (b) P(X ≤ 140) (c) P(|X − 150| ≥ 20) (d) P(135

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