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Fundamentals Of Probability With Stochastic Processes 4th Edition Saeed Ghahramani - Solutions

7. Prove that the function t/(1 − t), t < 1, cannot be the moment-generating function of a random variable.
6. Let X ∼ N(μ, σ2) and Y = 2X + 3. Find the distribution of Y .
3. Let X be a discrete random variable with the probability mass functionFind MX(t) and E(X). p(i) = 2(3), i=1,2,3,...; zero elsewhere.
2. Let X be a random variable with probability density function(a) Find MX(t), E(X), and Var(X).(b) UsingMX(t), calculate E(X).Hint: Note that by the definition of derivative, 1/4 if (-1,3) f(x) = otherwise.
Let the moment-generating function of a random variable X beSince the moment-generating function of a discrete random variable with the probability mass functionis MX(t) given previously, by Theorem 11.2, the probability mass function of X is p(i) ** + m + m + x = (1) * -
A positive random variable X is called lognormal with parameters μ andσ2 if lnX ∼ N(μ, σ2). Let X be a lognormal random variable with parameters μ and σ2.(a) For a positive integer r, calculate the rth moment of X.(b) Use the rth moment of X to find Var(X).(c) In 1977, a British researcher
Let Z be a standard normal random variable.(a) Calculate the moment-generating function of Z.(b) Use part (a) to find the moment-generating function of X, where X is a normal random variable with mean μ and variance σ2.(c) Use part (b) to calculate the mean and the variance of X.
Let X be an exponential random variable with parameter λ. Using momentgenerating functions, find E(Xn), where n is a positive integer
Let X be an exponential random variable with parameter λ. Using momentgenerating functions, calculate the mean and the variance of X.
Let X be a binomial random variable with parameters (n, p). Find the moment-generating function of X, and use it to calculate E(X) and Var(X).
Let X be a Bernoulli random variable with parameter p, that is,Determine MX(t) and E(Xn) 1-p if x 0 P(X=2)= P if x = 1 0 otherwise.
10. A fair die is tossed four times. Let X be the number of sixes obtained, and let Y be the length of the longest run of consecutive sixes. So if, for example, an outcome is N6N6;that is, a non-6, 6, a non-6, and 6, respectively, then X = 2, and Y = 1. Similarly, if an outcome is 66N6, then X = 3
9. Amber has decided to ask randomly selected people what their birthdays are until she has met people with birthdays on each and every day of the year. Ignoring February 29, find the expected value and the standard deviation of the number of people that Amber will need to meet in order to achieve
8. Hooroo Jackson, a professor of cinema studies, has 12 Blu-ray movies inside his briefcase.The subject of his current lesson is method acting, and he plans to show his students clips from each of the three Marlon Brando movies that he has in his briefcase. If Professor Jackson withdraws movies
7. Each month, the average number of claims received by an insurance company is a gamma random variable with parameters n and λ, n being a positive integer. If claims arrive according to a Poisson process, find the distribution of the total number of claims received by the insurance company during
6. A point (X, Y ) is selected randomly from the annulusThat is, the random point is inside the region between the concentric circles with radii 2 and 3. Show that X and Y are dependent but uncorrelated. R={(x,y): 4
5. A seismograph is an instrument that detects, measures, and accurately records the magnitude, depth, and location of an earthquake. Suppose that the lifetime of a certain seismograph, placed in the New Mexico desert, is exponential with mean 7 years. Furthermore, suppose that the seismograph will
4. A black and white (monochrome) CRT monitor uses a single electron gun to create images. A color monitor uses three electron guns (red, green, blue) to create the same quality images, but in color. Dale-Marie buys a monochrome CRT monitor, and Dan buys a color monitor both manufactured by the
3. For an integer n > 1, let X1, X2, . . . , Xn be identically distributed random variables with Var(Xi) = σ2, 1 ≤ i ≤ n. For i, j, 1 ≤ i 6= j ≤ n, let ρ(Xi,Xj) =a, for a constanta. Show that a ≥ −1 n − 1.Hint: Start with Var(X1 + X2 + · · · + Xn) ≥ 0.
2. A business has a benefit-limit policy to protect its finances from calamity losses; that is, payments for its indirect costs each time it closes down by a devastating calamity such as hurricane, tornado, flood, and snowstorm. According to this policy, each year, the insurance company pays the
1. At the office of a college provost, calls arrive at extension 1247 according to a Poisson processN1(t) : t ≥ 0with rate λ, and calls arrive at extension 1223 independently according to a Poisson processN2(t) : t ≥ 0with rate μ. Let N be the number of calls arriving at extension 1223
17. 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
16. Let {X1,X2,X3, . . .} be a sequence of independent and identically distributed exponential random variables with parameter λ. Let N be a geometric random variable with parameter p independent of {X1,X2,X3, . . .}. Find the distribution P function of N i=1 Xi.
15. Bus A arrives at a station at a random time between 10:00 A.M. and 10:30 A.M. tomorrow.Bus B arrives at the same station at a random time between 10:00 A.M. and the arrival time of bus A. Find the expected value of the arrival time of bus B.
14. Let the joint probability density function of X and Y be given by(a) Show that E(X) does not exist.(b) Find E(X|Y ). ye-y(1+2) f(x,y) = if x>0, y> 0 otherwise.
13. In terms of the means, variances, and the covariance of the random variables X and Y, find α and β for which E(Y − α − βX)2 is minimum. This is the method of least squares; it fits the “best” line y = α + βx to the distribution of Y .
12. Let the joint probability density function of X and Y be given by(a) Find the marginal probability density functions of X and Y .(b) Determine the correlation coefficient of X and Y . f(x, y) if 0 < y < x < elsewhere.
11. A random point (X, Y ) is selected from the rectangle [0, π/2] × [0, 1]. What is the probability that it lies below the curve y = sin x?
10. 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 .
9. Two dice are rolled. The sum of the outcomes is denoted by X and the absolute value of their difference by Y . Calculate the covariance ofX and Y . Are X and Y uncorrelated?Are they independent?
8. Let the joint probability mass function of discrete random variables X and Y be given byFind Cov(X, Y ). =(x + y) p(x,y) = if (x, y) = (1, 1), (1,3), (2, 3) otherwise.
7. Let X and Y be jointly distributed with ρ(X, Y ) = 2/3, σX = 1, Var(Y ) = 9. Find Var(3X − 5Y + 7).
6. Let the joint probability density function of X, Y, and Z be given byFind ρ(X, Y ), ρ(X,Z), and ρ(Y,Z). 8xyz f(x, y, z) = 0 if 0 <
5. Determine the expected number of tosses of a die required to obtain four consecutive sixes.
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 an = 1 + 1/2 + · · · + 1/n.
3. Let the joint probability density function of random variables X and Y beFind E(X2 + 2XY ). f(x,y): 3x3 + xy 3 if 0x1, 0 y 2 elsewhere.
2. Let the probability density function of a random variable X be given byFind E(X3 + 2X − 7). 2x-2 if 1 <
1. In a commencement ceremony, for the dean of a college to present the diplomas of the graduates, a clerk piles the diplomas in the order that the students will walk on the stage.However, the clerk mixes the last 10 diplomas in some random order accidentally. Find the expected number of the last
2. Let f(x, y) be a joint bivariate normal probability density function. Determine the point at which the maximum value of f is obtained.Hint: Note that f(x, y) is maximum if and only if Q(x, y) is minimum. Show that Q(x, y) is minimum if it is zero.
1. When a golf ball is dropped on concrete, the ratio of its speed after hitting the ground to its speed before hitting the ground is called its coefficient of restitution. Let X and Y be the coefficients of restitution of a certain brand of new golf balls and five-year-old golf balls,
8. Let the joint probability density function of random variables X and Y be bivariate normal. Show that if σX = σY , then X + Y and X − Y are independent random variables.Hint: Show that the joint probability density function ofX+Y andX−Y is bivariate normal with correlation coefficient 0.
7. Let Z andW be independent standard normal randomvariables. LetX and Y be defined bywhere σ1, σ2 > 0, −∞ Note: By this exercise, if the joint probability density function of X and Y is bivariate normal, X and Y can be written as sums of independent standard normal random variables. X =
6. Let the joint probability density function of two random variables X and Y be given byFind E(X | Y = y), E(Y | X = x), and ρ(X, Y ).Hint: To find ρ, use Lemma 10.3. 2 if 0 < y < x, 0 < x
5. Let f(x, y) be a joint bivariate normal probability density function. Determine the point at which the maximum value of f is obtained.
4. Let the joint probability density function of X and Y be bivariate normal. For what values of α is the variance of αX + Y minimum?
3. The joint probability density function of X and Y is bivariate normal with σX = σY =9, μX = μY = 0, and ρ = 0. Find (a) P(X ≤ 6, Y ≤ 12); (b) P(X2 + Y 2 ≤ 36).
2. Based on calculations of an actuary of an insurance company, the company’s auto insurance division provides an annual net underwriting income with mean and standard deviation, in millions of dollars, of 200 and 60, respectively. The corresponding dollar amounts for the company’s
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 bivariate normal with the following parameters: μX = 71, μY = 60, σX = 3, σY = 2.7, and ρ = 0.45. Find the probability that the height of the
At a certain university, the joint probability density function of X and Y, the grade point averages of a student in his or her freshman and senior years, respectively, is bivariate normal. From the grades of past years it is known that μX = 3, μY = 2.5, σX = 0.5,σY = 0.4, and ρ = 0.4. Find
2. At a certain location in downtown Chicago, there are ample parking spaces on the street.At that location, cars are parked illegally and independently of the police inspection times according to a Poisson process with rate λ. The duration of the time a car is illegally parked is exponentially
1. Niki is babysitting Hannah and Joshua all day today. Hannah cries at a Poisson rate of 10 times per day. Joshua cries at a Poisson rate of 7 times per day independently of Hannah.If it was Hannah who cried the last three times, what is the probability that the next baby who will cry is
27. Prove Theorem 10.8.
26. Let X and Y be two given random variables. Prove that Var(XY) EX|Y] - E(XY). =
25. Let X and Y be continuous random variables. Prove thatHint: Let Z = E(X|Y ). By conditioning on Y and using Example 10.21, first show that E(XZ) = E(Z2). E[(X E(X|Y))] = E(X) E[E(X\Y)].
24. Let X and Y be independent and identically distributed random variables. In terms of X and Y, find E(X | X + Y ).
23. For constants α and random variables X1, X2, and Y, show that (a) E(aXY) = aE(X |Y); (b) E(X1+X2Y) = E(X |Y) + E(X2 |Y).
22. Suppose that a device is powered by a battery. Since an uninterrupted supply of power is needed, the device has a spare battery. When the battery fails, the circuit is altered electronically to connect the spare battery and remove the failed battery from the circuit.The spare battery then
21. Recently, Larry taught his daughter Emily how to play backgammon. To encourage Emily to practice this game, Larry decides to play with her until she wins two of the recent three games. If the probability that Emily wins a game is 0.35 independently of all preceding and future games, find the
20. Each time that Steven calls his friend Adam, the probability that Adam is available to talk with him is p independently of other calls. On average, after how many calls has Steven not missed Adam k consecutive times?
19. During an academic year, the admissions office of a small college receives student applications at a Poisson rate of 5 per day. It is a policy of this college to double its student recruitment efforts if no applications arrive for two consecutive business days. Find the expected number of
18. In Rome, tourists arrive at a historical monument according to a Poisson process, on average, one every five minutes. There are guided tours that depart (a) whenever there is a group of 10 tourists waiting to take the tour, or (b) one hour has elapsed from the time the previous tour began. It
17. A fair coin is tossed successively. Let Kn be the number of tosses until n consecutive heads occur(c) By finding the expected values of both sides of (b) find a recursive relation between E(Kn) and E(Kn−1).(d) Note that E(K1) = 2. Use this and (c) to find E(Kn). (a) Argue that E(K, | Kn1 = i)
16. Suppose that X and Y represent the amount of money in the wallets of players A and B, respectively. Let X and Y be jointly uniformly distributed on the unit square[0, 1] × [0, 1]. A and B each places his wallet on the table. Whoever has the smallest amount of money in his wallet, wins all the
15. Two devices, the lifetime of which are independent exponential random variables with parameters λ and μ, are put on a life test. Suppose that the time until a catastrophe, such as a shock, that will fail both devices independently of their lifetimes is exponential with mean 1/γ. Find the
14. The lifetimes of batteries manufactured by a certain company are independent exponential random variables each with mean 1/λ. Andy loads his two-battery and his singlebattery flashlights with such batteries. What is the probability that he can shine his twobattery flashlight longer than his
13. At the intersection of two remote roads, the vehicles arriving are either cars or trucks.Suppose that cars arrive at the intersection at a Poisson rate of λ per minute, and trucks arrive at a Poisson rate of μ per minute. Suppose that the arrivals are independent of each other. If we are
12. At Berkeley, California, Lily commutes by bus every work day. To go from home to work, she arrives at the bus stop at the intersection of University and Shattuck Avenues at a random time between 8:30 A.M. and 8:45 A.M. She then takes bus number 67 or bus number 78, whichever arrives first.
11. Prove that, for a Poisson random variable N, if the parameter λ is not fixed and is itself an exponential random variable with parameter 1, then P(N = i) = = (-1) +1.
10. Suppose that X and Y are independent random variables with probability density functions f and g, respectively. Use conditioning technique to calculate P(X < Y ).
9. From an ordinary deck of 52 cards, cards are drawn at random, one by one and without replacement until a heart is drawn. What is the expected value of the number of cards drawn?Hint: Consider a deck of cards with 13 hearts and 39 − n nonheart cards. Let Xn be the number of cards to be drawn
8. In data communication, usually messages sent are combinations of characters, and each character consists of a number of bits. A bit is the smallest unit of information and is either 1 or 0. Suppose that the length of a character (in bits) is a geometric random variable with parameter p. Suppose
7. A typist, on average, makes three typing errors in every two pages. If pages with more than two errors must be retyped, on average how many pages must she type to prepare a report of 200 pages? Assume that the number of errors in a page is a Poisson random variable. Note that some of the retyped
6. The lifetime of a machine, in years, is a uniform random variable over the interval (0, 7).The machine will be replaced at age 5 or when it fails, if that occurs before age 5. Find the expected value of the age of the machine at the time of replacement.
5. If a car is under one year old and is totaled, an insurance company replaces that car with a new one. An actuary has calculated that when a $60,000-car gets involved in an accident, the probability of total loss is 0.015 and the probability of partial loss is 0.038.She has also approximated the
4. For given random variables Y and Z, letFind E(X) in terms of E(Y ) and E(Z). Y with probability p X = Z with probability 1 - p.
3. In a box, Lynn has b batteries of which d are dead. She tests them randomly and one by one. Every time that a good battery is drawn, she will return it to the box; every time that a dead battery is drawn, she will replace it by a good one.(a) Determine the expected value of the number of good
2. The orders received for grain by a farmer add up to X tons, where X is a continuous random variable uniformly distributed over the interval (4, 7). Every ton of grain sold brings a profit ofa, and every ton that is not sold is destroyed at a loss of a/3. How many tons of grain should the farmer
1. A fair coin is tossed until two tails occur successively. Find the expected number of the tosses required.Hint: Letand condition on X. X = 1 if the first toss results in tails if the first toss results in heads,
A fisherman catches fish in a large lake with lots of fish, at a Poisson rate of two per hour. If, on a given day, the fisherman spends randomly anywhere between 3 and 8 hours fishing, find the expected value and the variance of the number of fish he catches.
Let X be the natural lifetime of a device that also fails if a catastrophe such as a shock occurs. Let Y be the time until the next catastrophe. Suppose that X and Y are independent exponential random variables with parameters λ and μ, respectively. Given that X < Y, find the expected lifetime of
Suppose that Z1 and Z2 are independent standard normal random variables.Show that the ratio Z1/|Z2| is a Cauchy random variable. That is, Z1/|Z2| is a random variable with the probability density function 1 f(t) = -
The time between consecutive earthquakes in Los Angeles and the time between consecutive earthquakes in San Francisco are independent and exponentially distributed with means 1/λ and 1/μ, respectively.What is the probability that the next earthquake occurs in Los Angeles?
Suppose that the average number of breakdowns for a certain airplane is 12.5 times a year. If the expected value of repair time is 7 days for each breakdown, and if the repair times are identically distributed, independent random variables, find the expected total repair time. Assume that repair
What is the expected number of random digits that should be generated to obtain three consecutive zeros?
Let X and Y be continuous random variables with joint probability density functionFind E(X|Y ). (x + y) if 01, 0
Suppose that N(t), the number of people who pass by a museum at or prior to t, is a Poisson process having rate λ. If a person passing by enters the museum with probability p, what is the expected number of people who enter the museum at or prior to t?
2. For random variables X1, X2, . . . , Xn, we have that Var(Xi) = 4, 1 ≤ i ≤ n, and for 1 ≤ i 6= j ≤ n, ρ(Xi,Xj) = −1/16. Find Var(X1 + X2 + · · · + Xn).Hint: Note that Pn−1 i=1 Pn j=i+1 Cov(Xi,Xj) has (n − 1) + (n − 2) + · · · + 1 =[(n − 1)n]/2 terms.
1. In a probability exam, for two random variables X and Y with a given joint probability density function, Fiona’s calculations resulted in E(X) = E(Y ) = 2, E(X2) = 13, E(Y 2) = 40, and E(XY ) = 23. However, Fiona got no points for these answers.Under her solution, the professor wrote,
7. Show that if the joint probability density function of X and Y isthen there exists no linear relation between X and Y . sin(x+y) if 0x < f(x,y): elsewhere, 67 0 y 2
6. Prove that if Cov(X, Y ) = 0, then Var(X) Var(Y) p(X+Y,XY) =: Var(X) + Var(Y)"
4. For real numbers α and β, letProve that for random variables X and Y, sgn(a)=0 1 if a > 0 if a 0 -1 if a < 0.
3. A stick of length 1 is broken into two pieces at a random point. Find the correlation coefficient and the covariance of these pieces.
2. Let the joint probability density function of X and Y be given byCalculate the correlation coefficient of X and Y . sin r sin y if 0/2, 0 y /2 f(x,y) = 0 otherwise.
Show that if X and Y are continuous random variables with the joint probability density functionthen X and Y are not linearly related f(x, y) = = (x+y if 0 <
2. Let X and Y be independent and identically distributed exponential random variables with parameter λ. Let U = max(X, Y ) and V = min(X, Y ). Using the relation Cov(U, V ) = Cov(U − V + V, V ) = Cov(U − V, V ) + Cov(V, V ), Calculate Cov(U, V ) in terms of Var(V ).
1. We draw 8 cards, one at a time, randomly, and without replacement from an ordinary deck of 52 cards. LetFor 1 ≤ i if the ith card drawn is a heart X otherwise.
26. Exactly n married couples are living in a small town.What is the variance of the surviving couples afterm deaths occur among them? Assume that the deaths occur at random, there are no divorces, and there are no new marriages.Note: This situation involves the Daniel Bernoulli problem discussed
25. Let X be a hypergeometric random variable with probability mass functionRecall that X is the number of defective items among n items drawn randomly and without replacement from a box containingD defective and N −D nondefective items.Show thatHint: LetAi be the event that the ith item drawn is
24. Show that if X1,X2, . . . ,Xn are random variables and a1, a2, . . . , an are constants, then n Var( a;X;) = a Var(X;) +2 a;a; Cov(X, X;). i=1 i=1 i
23. A fair die is thrown n times.What is the covariance of the number of 1’s and the number of 6’s obtained?Hint: Use the result of Exercise 22.

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