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probability and stochastic modeling

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

9. Let X be a continuous random variable with set of possible values {x: 0 f. Using integration by parts, prove the following special case of Theorem 6.2. E(X) = [1 = [1-F(t)] dt. 0
8. Prove or disprove: If Pn i=1 αi = 1, αi ≥ 0, ∀i, andfi ni=1 is a sequence of probability density functions, then Pn i=1 αifi is a probability density function
7. The probability density function of a continuous random variable X isFind the probability density function of Y = X4. (30x2 (1-x)2 if 0x1 f(x)= 0 otherwise.
6. Let X be a random variable with probability density functionFind the probability density functions of Y = eX, Z = X2, andW = (X − 1)2. f(x) = (43/15 1x2 otherwise.
5. Does there exist a constant c for which the following is a probability density function? C if x > 0 f(x)= 1+2 0 otherwise.
4. Let X be a random variable with probability density functionFind P(−2 f(x)= e-11 2
3. Let X be a continuous random variable with probability density function f(x) = 6x(1 − x), 0 < x < 1.What is the probability that X is within two standard deviations of the mean?
2. Let X be a continuous random variable with the probability density functionFind E(X) and Var(X) if they exist. (2/23 if x 1 f(x) = otherwise.
1. Let X be a random number from (0, 1). Find the probability density function of the random variable Y = 1/X.
23. Let X be a continuous random variable with probability density functionf. Show that if E(X) exists; that is, ifthen f(x) dx
22. Suppose that X is the lifetime of a randomly selected fan used in certain types of diesel engines. Let Y be a randomly selected competing fan for the same type of diesel engines manufactured by another company. To compare the lifetimes X and Y, it is not sufficient to compare E(X) and E(Y ).
21. Let X be the random variable introduced in Exercise 14. Applying the results of Exercise 19, calculate Var(X).
20. Let X be a continuous random variable. Prove thatThese important inequalities show that if and only if the seriesHint: By Exercise 19,Note that on the interval [n, n + 1), n=1 P(Xn) E(|X|) 1+ P(|X| n). =1
19. Let X be a nonnegative random variable with distribution function F. Define(a) Prove that R ∞0 I(t) dt = X.(b) By calculating the expected value of both sides of part (a), prove thatThis is a special case of Theorem 6.2.(c) For r > 0, use part (b) to prove that I(t)= if X > t otherwise.
18. Let X be a continuous random variable with probability density function f(x). Determine the value of y for which E????|X − y|is minimum.
17. Let X be a continuous random variable with probability density functionf. A number t is said to be the median of X ifBy Exercise 9, Section 6.1, X is symmetric about α if and only if for all x we have f(α − x) = f(α + x). Show that if X is symmetric about α, then E(X) = Median(X) = α.
16. Let X be a continuous random variable with the probability density functionProve that - sin x x if 0
15. For n ≥ 1, let Xn be a continuous random variable with the probability density functionXn’s are called Pareto random variables and are used to study income distributions.(a) Calculate cn, n ≥ 1.(b) Find E(Xn), n ≥ 1.(c) Determine the probability density function of Zn = lnXn, n ≥
14. Suppose that X, the interarrival time between two customers entering a certain postoffice, satisfieswhere α + β = 1, α ≥ 0, β ≥ 0, λ > 0, μ > 0. Calculate the expected value of X.Hint: For a fast calculation, use Remark 6.4. P(X >t)=ae + Bet -Xt t 0,
Let X be a random variable with the probability density functionProve that E ????|X|αconverges if 0 f(x) = 1 (1+x)' -x < x < .
12. Let X be a random variable with probability density functionCalculate Var(X). f(x)= |2 -x < x < .
11. A right triangle has a hypotenuse of length 9. If the probability density function of one side’s length is given bywhat is the expected value of the length of the other side? [x/6 if 2
10. Find E(lnX) if X is a continuous random variable with probability density function f(x)= [2/22 if 1 < x < 2 elsewhere.
9. Let the probability density function of tomorrow’s Celsius temperature be h. In terms of h, calculate the corresponding probability density function and its expected value for Fahrenheit temperature.Hint: Let C and F be tomorrow’s temperature in Celsius and Fahrenheit, respectively.Then F =
8. Let Y be a continuous random variable with distribution functionwhere A, k, and α are positive constants. (Such distribution functions arise in the study of local computer network performance.) Find E(Y ). e-k(a-y)/A -x , F(y)= 1
7. Find the expected value of a random variable X with the probability density function f(x)= 0 1 if-1 <
6. A random variable X has the probability density functionCalculate E(eX). 3e-3x if 0 < x < f(x) = 3e- otherwise.
4. At a university, which does not qualify as a tax-exempt entity under federal and state laws, each year, without taxes, it costs an average of $56,000 with a variance of 81 million dollars to repair and maintain all the dormitories. If all items associated with the maintenance and repair of the
1. The distribution function for the duration of a certain soap opera (in tens of hours) is(a) Find E(X).(b) Show that Var(X) does not exist. 16 if x 4 F(x) 22 if x < 4.
The time elapsed, in minutes, between the placement of an order of pizza and its delivery is random with the probability density function(a) Determine the mean and standard deviation of the time it takes for the pizza shop to deliver pizza.(b) Suppose that it takes 12 minutes for the pizza shop to
A point X is selected from the interval (0, π/4) randomly. Calculate E(cos 2X) and E(cos2 X).
A random variable X with probability density functionis called a Cauchy random variable.(a) Find c.(b) Show that E(X) does not exist. f(x) = 1+x21
In a group of adult males, the difference between the uric acid value and 6, the standard value, is a random variable X with the following probability density functionCalculate the mean of these differences for the group 27 (3x-2x) if 2/3
2. Let X be a random variable with probability density functionUsing the method of transformations, find the probability density function of Y = X2.Then calculate E(Y ). f(x) = (re-12/2 if x 0 0 otherwise.
1. Let X be a random point from the interval (−π/2, π/2). Find the probability density function of Y = tanX.
9. Let X be a random variable with the probability density function given byLetFind the probability density function of Y . f(x) = Se- { 0 if x 0 elsewhere.
8. Let X be a random variable with the probability density function(X is called a Cauchy random variable.) Find the probability density function of Z = arctanX. f(x) = 1 (1+x)'
7. Let f be the probability density function of a randomvariableX. In terms off, calculate the probability density function of X2.
6. Let the probability density function of X befor some λ > 0. Using the method of distribution functions, calculate the probability density function of Y = 3√ X2. Ae-Ar f(x) = if x 0 0 otherwise,
5. Let X be a continuous random variable with the probability density functionUsing the method of transformations, find the probability density function of the random variable Y = log2 X. 3e-32 f(x) if x > 0 otherwise.
4. Let the probability density function of X beUsing the method of transformations, find the probability density functions of Y = X√X and Z = e−X. f(x): 10 if x > 0 elsewhere.
3. Let X be a continuous random variable with distribution function F and probability density functionf. Calculate the probability density function of the random variable Y = eX.
2. A manufacturer produces metal disks with radii being random variables with the identical probability density functionWhat is the probability that the area of a random disk is greater than 39? 2:5 5x4 if 3x4 f(x) = 431 2586 otherwise.
Let X be a continuous random variable with the probability density functionUsing the method of transformations, find the probability density function of the random variable Y = 1 − 3X2. 423 if 0 < x
Let X be a random variable with the probability density functionUsing the method of transformations, find the probability density function of Y = √X. 2e-2x fx(x) = if x > 0 0 otherwise.
The error of a measurement has the probability density functionFind the distribution and the probability density functions of the magnitude of the error. [1/2 f(x) = 0 if-1 <
LetX be a continuous random variable with distribution function F and probability density functionf. In terms off, find the distribution and the probability density functions of Y = X3.
Let X be a continuous random variable with the probability density functionFind the distribution and the probability density functions of Y = X2. [2/x2 if 1 < x
2. The distribution for the duration of a certain soap opera, in tens of hours, isIf the soap opera has been running for 110 hours, what is the probability that it will run for at least another 110 hours? 16 x 4 F(x)= 22 0 otherwise.
1. For what value of α is the following the probability density function of a randomvariable X? For that value of α, find P(1.5 (9/24 1 <
15. The distribution function of a random variable X is given byDetermine the constants α and β and the probability density function of X. F(x) = a+Barctan -x < x < .
13. Let X be a continuous random variable with probability density functionFind the distribution function of X. 1-|x| -1 <
11. Suppose that the loss in a certain investment, in thousands of dollars, is a continuous random variable X that has a probability density function of the form(a) Calculate the value of k.(b) Find the probability that the loss is at most $500. f(x) = (k(2x-3x) -1
10. The functionis continuous, differentiable, nondecreasing, F(−∞) = 0, and F(∞) = 1. Is F the distribution function of a continuous random variable? F(x)= 0 { COS I 1 x
9. Let X be a continuous random variable with probability density function f.We say that X is symmetric about if for all x,(a) Prove that X is symmetric about α if and only if for all x, we have f(α − x) = f(α + x).(b) Show that X is symmetric about α if and only if f(x) = f(2α − x) for
8. An insurance company has sold a large number of homeowner insurance policies in an affluent area, where the insured value of homes is above $750,000. An actuary has calculated that X, the insured value of a randomly selected home, in millions of dollars, is a continuous random variable with
6. At a grocery store in Idaho, the demand for honey, in kilograms, per week, is a random variable with probability density functionHow many kilograms of honey must the grocery store carry per week so that the stock out probability for honey is at most 0.05? f(x) = 5 6656 (4-x)(4+x) 0 < x
5. The probability density function of a random variable X is given by(a) Calculate the value of c.(b) Find the distribution function of X. f(x)=1x2 if-1 <
4. The lifetime of a tire selected randomly from a used tire shop is 10, 000X miles, where X is a random variable with the probability density function(a) What percentage of the tires of this shop last fewer than 15,000 miles?(b) What percentage of those having lifetimes fewer than 15,000 miles
3. The time it takes for a student to finish an aptitude test (in hours) has a probabilitydensity function of the form(a) Determine the constant c.(b) Calculate the distribution function of the time it takes for a randomly selected student to finish the aptitude test.(c) What is the probability
2. The distribution function for the duration of a certain soap opera (in tens of hours) is(a) Calculatef, the probability density function of the soap opera.(b) Sketch the graphs of F and f.(c) What is the probability that the soap opera takes at most 50 hours? At least 60 hours? Between 50 and 70
1. When a certain car breaks down, the time that it takes to fix it (in hours) is a random variable with the probability density function(a) Calculate the value of c.(b) Find the probability that when this car breaks down, it takes at most 30 minutes to fix it. f(x)= -for -3x ce if 0 < x <
(a) Sketch the graph of the functionand show that it is the probability density function of a random variable X.(b) Find F, the distribution function of X, and sketch its graph.(c) Show that F is continuous. | x - 3| 1 x 5 f(x)=2 otherwise,
Experience has shown that while walking in a certain park, the time X, in minutes, between seeing two people smoking has a probability density function of the form(a) Calculate the value of λ.(b) Find the distribution function of X.(c) What is the probability that Jeff, who has just seen a person
10. A company has insurance coverage to pay for its indirect cost each time it closes down from a devastating calamity such as hurricane, tornado, flood, and snowstorm. Suppose that the policy pays $50,000 for each calamity that causes the company to shut down temporarily, except for the first
9. A gambler has two dice, an unbiased one, and a loaded one which, when tossed, lands on 6 with probability of 4/9 and lands on any of the other faces with probability of 1/9.The gambler does not know which die is the loaded one. He chooses one of the two dice at random and tosses that die ten
8. Suppose that, independently of other tornado-force winds, with probability 0.063, such a wind even damages tornado-proof fortified rooms that are constructed with cinder blocks and are filled with mortar and rebar. Such a room is constructed in a town in the plains located in a Tornado alley. If
7. To reward improved product quality, a small manufacturing company that has only 27 employees, has developed a cash incentive program. Every year, the company pays out a $7000 bonus on top of the salaries of the individuals who have performed at or above certain thresholds. Suppose that,
6. There are two fair coins on a table. The first coin is flipped successively and independently until a heads appears. Then the second coin is flipped repeatedly and independently until a heads appears. Let X be the number of flips until the first coin lands heads, and let Y be the number of flips
5. To estimate the number of kangaroos living in a particular region of Western Victoria in Australia, Liam captured 25 kangaroos, marked them and then let them go free. After a few days he captured 30 kangaroos and observed that 5 of them were marked. Using this experiment, estimate n, the total
4. Ginny has begun a new website that aggregates reviews of movies and TV shows. Suppose that, all over the world, every day, 10 million people search for film review websites, and Ginny’s site is one of the last sites out of hundreds that a search turns up.If independently of the others, the
3. For certain software, independently of other users, the probability is 0.07 that a user encounters a fault. Let X be the number of users who do not encounter a fault before the 12th user who encounters a fault. Find the probability mass function of X.Hint: Note that X + 12 is a negative binomial
2. A company produces liquid silicone rubber that is designed specifically for mold making.If in the molding process, defective items are produced at a rate of 0.6%, find the probability that, in a sample of 200 items, there are at most 3 defective ones.
28. Show that if all three of n, N, and D → ∞ so that n/N → 0, D/N converges to a small number, and nD/N → λ, then for all x,This formula shows that the Poisson distribution is the limit of the hypergeometric distribution. N-D n-x I N n x!
26. Suppose that n babies were born at a county hospital last week. Also suppose that the probability of a baby having blonde hair is p. If k of these n babies are blondes, what is the probability that the ith baby born is blonde?
24. A bowl contains w white and b blue chips. Chips are drawn at random and with replacement until a blue chip is drawn. What is the probability that (a) exactly n draws are required; (b) at least n draws are required?
23. The probability that a child of a certain family inherits a certain disease is 0.23 independently of other children inheriting the disease. If the family has five children and the disease is detected in one child, what is the probability that exactly two more children have the disease as well?
22. From the set {x: 0 ≤ x ≤ 1} numbers are selected at random and independently and rounded to three decimal places. What is the probability that 0.345 is obtained (a) for the first time on the 1000th selections; (b) for the third time on the 3000th selections?
20. Passengers are making reservations for a particular flight on a small commuter plane 24 hours a day at a Poisson rate of 3 reservations per 8 hours. If 24 seats are available for the flight, what is the probability that by the end of the second day all the plane seats are reserved?
19. At the Antonio Car dealership, the probability that during a given month at least one consumer returns a car for warranty work is 0.45. Suppose that the numbers of cars returned for warranty work in different months are independent. Consider the period until the fifth month in which at least
3. A restaurant serves 8 fish entr´ees, 12 beef, and 10 poultry. If customers select from these entr´ees randomly, what is the expected number of fish entr´ees ordered by the next four customers?
2. At the Antonio Car dealership, the probability that during a month at least one consumer returns a car for warranty work is 0.45. Suppose that the numbers of cars returned for warranty work in different months are independent. Consider the period until the fifth month in which at least one car
1. A maximum of m (m ≥ 1) independent Bernoulli trials, each with probability of success p, 0 < p < 1, are performed successively. However, we stop the trials if the first success occurs before the mth trial. Let Z be the number of experiments performed.Find the probability mass function of Z.
32. Suppose that, from a box containing D defective and N − D nondefective items, n(n ≤ D) are drawn one by one, at random and without replacement.(a) Find the probability that the kth item drawn is defective.(b) If the (k − 1)st item drawn is defective, what is the probability that the kth
31. To estimate the number of trout in a lake, we caught 50 trout, tagged and returned them.Later we caught 50 trout and found that four of them were tagged. From this experiment estimate n, the total number of trout in the lake.Hint: Let pn be the probability of four tagged trout among the 50
29. In the Banach matchbox problem, Example 5.25, find the probability that the box which is emptied first is not the one that is first found empty.
28. In the Banach matchbox problem, Example 5.25, find the probability that when the first matchbox is emptied (not found empty) there are exactly m matches in the other box.
27. Twelve hundred eggs, of which 200 are rotten, are distributed randomly in 100 cartons, each containing a dozen eggs. These cartons are then sold to a restaurant. How many cartons should we expect the chef of the restaurant to open before finding one without rotten eggs?
26. In data communication,messages are usually 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. Furthermore,
25. Let X be a geometric random variable with parameter p, and n and m be nonnegative integers.(a) For what values of n is P(X = n) maximum?(b) What is the probability that X is even?(c) Show that the geometric is the only distribution on the positive integers with the memoryless property:P(X > n
24. On average, how many independent games of poker are required until a preassigned player is dealt a straight? (See Exercise 35 of Section 2.4 for a definition of a straight.The cards have distinct consecutive values that are not of the same suit: for example, 3 of hearts, 4 of hearts, 5 of
23. A fair coin is flipped repeatedly. What is the probability that the fifth tail occurs before the tenth head?
22. A computer network consists of several stations connected by various media (usually cables). There are certain instances when no message is being transmitted. At such “suitable instances,” each station will send a message with probability p, independently of the other stations. However, if
20. The probability is p that a message sent over a communication channel is garbled. If the message is sent repeatedly until it is transmitted reliably, and if each time it takes 2 minutes to process it, what is the probability that the transmission of a message takes more than t minutes?
18. Let X be a geometric random variable with parameter p, 0 < p < 1. Find E 1 1 + X.
16. Suppose that 15% of the population of a town are senior citizens. Let X be the number of nonsenior citizens who enter a mall before the tenth senior citizen arrives. Find the probability mass function of X. Assume that each customer who enters the mall is a random person from the entire
15. In recent months, an electric utility company has received complaints from its customers who think that, due to their unusually high electricity bills, their electric meters are defective. The company begins to inspect the meters one by one. If only 3.6% of them are defective, what is the
14. For certain software, independently of other users, the probability is 0.07 that a user encounters a fault.What are the chances that the 30th user is the 5th person encountering a fault?
5. Suppose that 20% of a group of people have hazel eyes. What is the probability that the eighth passenger boarding a plane is the third one having hazel eyes? Assume that passengers boarding the plane form a randomly chosen group.
4. The probability is p that Marty hits targetM when he fires at it. The probability is q that Alvie hits target A when he fires at it. Marty and Alvie fire one shot each at their targets.If both of them hit their targets, they stop; otherwise, they will continue.(a) What is the probability that

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