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Introduction To Mathematical Statistics And Its Applications 5th Edition Richard J. Larsen, Morris L. Marx - Solutions

For one-pair poker hands, why is the number of denominations for the three single cards(12/3) rather than (12/1)(11/1)(10/1)?
Dana is not the world's best poker player. Dealt a 2 of diamonds, an 8 of diamonds, an ace of hearts, an ace of clubs, and an ace of spades, she discards the three aces. What are her chances of drawing to a flush?
Timis dealt a 4 of clubs, a 6 of hearts, an 8 of hearts, a 9 of hearts, and a king of diamonds. He discards the 4 and the king. What are his chances of drawing to a straight flush? To a flush?
An urn contains six chips, numbered 1 through 6. Two are chosen at random and their numbers are added together. What is the probability that the resulting sum is equal to 5?
Five cards are dealt from a standard 52-card deck. What is the probability that the sum of the faces on the five cards is 48 or more?
A somewhat inebriated conventioneer finds himself in the embarrassing predicament of being unable to predetermine whether his next step will be forward or backward. What is the probability that after hazarding n such maneuvers he will have stumbled forward a distance of r steps? (Let x denote the
An urn contains twenty chips, numbered 1 through 20. Two are drawn simultaneously. What is the probability that the numbers on the two chips will differ by more than 2?
A bridge hand (thirteen cards) is dealt from a standard 52-card deck. Let A be the event that the hand contains four aces; let B be the event that the hand contains four kings. Find P(A ∪ B).
Consider a set of ten urns, nine of which contain three white chips and three red chips each. The tenth contains five white chips and one red chip. An urn is picked at random. Then a sample of size 3 is drawn without replacement from that urn. If all three chips drawn are white, what is the
Five fair dice are rolled. What is the probability that the faces showing constitute a "full house"-that is, three faces show one number and two faces show a second number?
Imagine that the test tube pictured contains 2n grains of sand, n white and n black. Suppose the tube is vigorously shaken. What is the probability that the two colors of sand will completely separate; that is, all of one color fall to the bottom, and all of the other color lie on top?
Suppose the length of time, in minutes, that you have to wait at a bank teller’s window is uniformly distributed over the interval (0, 10). If you go to the bank four times during the next month, what is the probability that your second longest wait will be less than five minutes?
Suppose that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the entire sample?
In a certain large metropolitan area, the proportion, Y, of students bused varies widely from school to school. The distribution of proportions is roughly described by the following pdf:Suppose the enrollment figures for five schools selected at random are examined. What is the probability that the
Consider a system containing n components, where the lifetimes of the components are independent random variables and each has pdf fY(y) = λe−λy, y > 0. Show that the average time elapsing before the first component failure occurs is 1/nλ.
Let Y1, Y2, . . ., Yn be a random sample from a uniform pdf over [0, 1]. Use Theorem 3.10.2 to show that
Use Question 3.10.13 to find the expected value of Yʹi, where Y1, Y2, . . ., Yn is a random sample from a uniform pdf defined over the interval [0, 1].
Suppose three points are picked randomly from the unit interval. What is the probability that the three are within a half unit of one another?
Suppose a device has three independent components, all of whose lifetimes (in months) are modeled by the exponential pdf, fY(y) = e−y, y > 0. What is the probability that all three components will fail within two months of one another?
A random sample of size n = 6 is taken from the pdf fY(y) = 3y2, 0 ≤ y ≤ 1. Find P(Yʹ5 > 0.75).
What is the probability that the larger of two random observations drawn from any continuous pdf will exceed the sixtieth percentile?
A random sample of size 5 is drawn from the pdf fY(y) = 2y, 0 ≤ y ≤ 1. Calculate P(Yʹ1 < 0.6 < Yʹ5).
Suppose that Y1, Y2, . . ., Yn is a random sample of size n drawn from a continuous pdf, fY(y), whose median is m. Is P(Yʹ1 > m) less than, equal to, or greater than P(Yʹn > m)?
Let Y1, Y2, . . ., Yn be a random sample from the exponential pdf fy(y) = e−y, y ≥ 0. What is the smallest n for which P(Ymin < 0.2) > 0.9?
Calculate P(0.6 < Yʹ4 < 0.7) if a random sample of size 6 is drawn from the uniform pdf defined over the interval [0, 1].
A random sample of size n =5 is drawn from the pdf fY(y) = 2y, 0 ≤ y ≤ 1. On the same set of axes, graph the pdfs for Y2, Yʹ1, and Yʹ5.
Suppose that n observations are taken at random from the pdfWhat is the probability that the smallest observation is larger than twenty?
Suppose X and Y have the joint pdf pX,Y(x, y) = x + y + xy/21 for the points (1, 1), (1, 2), (2, 1), (2, 2), where X denotes a "message" sent (either x = 1 or x = 2) and Y denotes a "message" received. Find the probability that the message sent was the message received-that is, find pY|x(y).
Suppose Compositor A is preparing a manuscript to be published. Assume that she makes X errors on a given page, where X has the Poisson pdf, pX(k) = e−22k/k!, k = 0, 1, 2, . . . . A second compositor, B, is also working on the book. He makes Y errors on a page, where pY (k) = e−33k/k!, k = 0,
Let X be a nonnegative random variable. We say that X is memory less ifP(X > s + t|X > t) = P(X > s) for all s, t ≥ 0Show that a random variable with pdf fX(x) = (1/λ)e−x/λ, x > 0, is memory less.
Given the joint pdf fX,Y(x, y) = 2e−(x + y), 0 ≤ x ≤ y, y ≥ 0 find (a) P(Y < 1|X < 1). (b) P(Y < 1|X = 1). (c) fY|x(y). (d) E(Y|x).
Find the conditional pdf of Y given x if fX,Y(x, y) = x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.
If fX,Y(x, y) = 2, x ≥ 0, y ≥ 0, x + y ≤ 1 show that the conditional pdf of Y given x is uniform.
Suppose that fY|x(y) = 2y + 4x/1 + 4x and fX(x) = 1/3・ (1 + 4x) for 0 < x < 1 and 0 < y < 1. Find the marginal pdf for Y.
Suppose that X and Y are distributed according to the joint pdf fX,Y(x, y) = 2/5・ (2x + 3y), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 Find (a) fX(x). (b) fY|x(y). (c) P(1/4 ≤ Y ≤ 3/4|X = 1/2.
If X and Y have the joint pdf fX,Y(x, y) =2, 0 ≤ x < y ≤ 1 find P(0 < X < 1/2|Y = 3/4).
Find P(X < 1|Y = 1 1/2) if X and Y have the joint Pdf fX,Y(x, y) = xy/2, 0 ≤ x < y ≤ 2
Suppose that X1, X2, X3, X4, and X5 have the joint pdf fX1,X2,X3,X4,X5(x1, x2, x3, x4, x5) = 32x1x2x3x4x5 for 0 < xi < 1, i = 1, 2, . . . , 5. Find the joint conditional pdf of x1, x2, and x3 given that x4 = x4 and x5 = x5.
Suppose a die is rolled six times. Let X be the total number of 4's that occur and let Y be the number of 4's in the first two tosses. Find pY|x(y).
Suppose the random variables X and Y are jointly distributed according to the Pdf fX,Y(x, y) = 6/7 (x2 + xy/2), 0 ≤ x ≤ 1, 0 ≤ y ≤ 2 Find (a) fX(x). (b) P(X > 2Y). (c) P(Y > 1|X > 1/2).
An urn contains eight red chips, six white chips, and four blue chips. A sample of size 3 is drawn without replacement. Let X denote the number of red chips in the sample and Y, the number of white chips. Find an expression for pY|x(y).
Five cards are dealt from a standard poker deck. Let X be the number of aces received, and Y the number of kings. Compute P(X = 2|Y = 2).
Given that two discrete random variables X and Y follow the joint pdf pX,Y(x, y) = k(x + y), for x = 1, 2, 3 and y = 1, 2, 3, (a) Find k. (b) Evaluate pY|x(1) for all values of x for which px(x) > 0.
Let X denote the number on a chip drawn at random from an urn containing three chips, numbered 1, 2, and 3. Let Y be the number of heads that occur when a fair coin is tossed X times. (a) Find pX,Y(x, y). (b) Find the marginal pdf of Y by summing out the x values.
Suppose X, Y, and Z have a trivariate distribution described by the joint pdf pX,Y,Z(x, y, z) = xy + xz + yz/54 where x, y, and z can be 1 or 2. Tabulate the joint conditional pdf of X and Y given each of the two values of z.
In Question 3.11.7 define the random variable W to be the "majority" of x, y, and z. For example, W(2, 2, 1) = 2 and W(1, 1, 1) = 1. Find the pdf of W|x.
Let X and Y be independent random variables where px (k) = e−λ λk/k! and pY(k) = e−μ μk/k! for k = 0, 1, . . . . Show that the conditional pdf of X given that X + Y = n is binomial with parameters n and λ/λ + μ.
Let X be a random variable with pdf pX (k) = 1/n, for k = 0, 1, 2, . . . , n − 1 and 0 otherwise. Show that MX(t) = 1−ent/n(1 − et).
Find E(Y4) if Y is an exponential random variable with fY(y) = λe−λy, y > 0.
The form of the moment-generating function for a normal random variable is MY(t) = eat+b2t2/2 (recall Example 3.12.4). Differentiate MY(t) to verify that a = E(Y) and b2 = Var(Y).
What is E(Y4) if the random variable Y has moment-generating function MY(t) = (1 − αt)−k?
Find E(Y2) if the moment-generating function for Y is given by MY(t) = e−t+4t2. Use Example 3.12.4 to find E(Y2) without taking any derivatives.
Find an expression for E(Yk) if MY(t) = (1 − t/λ)−r, where λ is any positive real number and r is a positive integer.
Use MY(t) to find the expected value of the uniform random variable described in Question 3.12.1.
Find the variance of Y if MY(t) = e2t /(1−t2).
Use Theorem 3.12.3(a) and Question 3.12.8 to find the moment-generating function of the random variable Y, where fY(y) = λye−λy, y ≥ 0.
Let Y1, Y2, and Y3 be independent random variables, each having the pdf of Question 3.12.17. Use Theorem 3.12.3(b) to find the moment-generating function of Y1 + Y2 + Y3. Compare your answer to the moment generating function in Question 3.12.14.
Use Theorems 3.12.2 and 3.12.3 to determine which of the following statements is true:(a) The sum of two independent Poisson random variables has a Poisson distribution.(b) The sum of two independent exponential random variables has an exponential distribution.(c) The sum of two independent normal
Two chips are drawn at random and without replacement from an urn that contains five chips, numbered 1 through 5. If the sum of the chips drawn is even, the random variable X equals 5; if the sum of the chips drawn is odd, X = −3. Find the moment-generating function for X.
Calculate P(X ≤ 2) if MX(t) = (1/4 + 3/4 et)5.
Suppose that Y1, Y2, . . ., Yn is a random sample of size n from a normal distribution with mean Î¼ and standard deviation Ïƒ. Use moment-generating functions to deduce the pdf of
Suppose the moment-generating function for a random variable W is given byCalculate P(W ‰¤ 1).
Suppose that X is a Poisson random variable, where pX(k) = e−λλk/k!, k = 0, 1, . . . . (a) Does the random variable W = 3X have a Poisson distribution? (b) Does the random variable W =3X +1 have a Poisson distribution?
Suppose that Y is a normal variable, where(a) Does the random variable W = 3Y have a normal distribution?(b) Does the random variable W = 3Y + 1 have a normal distribution?
Find the expected value of e3X if X is a binominal random variable with n =10 and p = 1/3.
Find the moment-generating function for the discrete random variable X whose probability function is given by pX(k) = (3/4)k (1/4), k = 0, 1, 2, . . .
Which pdfs would have the following moment generating functions? (a) MY(t) = e6t2 (b) MY(t) = 2/(2 − t) (c) MX(t) = (1/2 + 1/2 et)4 (d) MX(t) = 0.3et/(1 − 0.7et)
Let Y have pdfFind MY (t).
A random variable X is said to have a Poisson distribution if pX(k) = P(X = k) = eˆ’Î»Î»k/k!, k = 0, 1, 2, . . . . Find the moment-generating function for a Poisson random variable. Recall that
Let Y be a continuous random variable with fY(y) = ye−y, 0 ≤ y. Show that MY(t) = 1/(1 − t)2.
Calculate E(Y3) for a random variable whose moment-generating function is MY (t) = et2/2.
An investment analyst has tracked a certain blue chip stock for the past six months and found that on any given day, it either goes up a point or goes down a point. Furthermore, it went up on 25% of the days and down on 75%. What is the probability that at the close of trading four days from now,
The gunner on a small assault boat fires six missiles at an attacking plane. Each has a 20% chance of being on-target. If two or more of the shells find their mark, the plane will crash. At the same time, the pilot of the plane fires ten air-to-surface rockets, each of which has a 0.05 chance of
If a family has four children, is it more likely they will have two boys and two girls or three of one sex and one of the other? Assume that the probability of a child being a boy is 1 2 and that the births are independent events.
Experience has shown that only 1/3 of all patients having a certain disease will recover if given the standard treatment. A new drug is to be tested on a group of twelve volunteers. If the FDA requires that at least seven of these patients recover before it will license the new drug, what is the
Transportation to school for a rural county’s seventy-six children is provided by a fleet of four buses. Drivers are chosen on a day-to-day basis and come from a pool of local farmers who have agreed to be “on call.” What is the smallest number of drivers who need to be in the pool if the
The captain of a Navy gunboat orders a volley of twenty-five missiles to be fired at random along a five-hundred-foot stretch of shoreline that he hopes to establish as a beachhead. Dug into the beach is a thirty-foot-long bunker serving as the enemy’s first line of defense. The captain has
A computer has generated seven random numbers over the interval 0 to 1. Is it more likely that (1) Exactly three will be in the interval 1/2 to 1 or (2) Fewer than three will be greater than 3/4?
Listed in the following table is the length distribution of World Series competition for the 58 series from 1950 to 2008 (there was no series in 1994).World Series LengthsNumber of Games, X Number of
Use the expansion of (x + y) n (recall the comment in Section 2.6 on p. 67) to verify that the binomial probabilities sum to 1; that is,
Suppose a series of n independent trials can end in one of three possible outcomes. Let k1 and k2 denote the number of trials that result in outcomes 1 and 2, respectively. Let p1 and p2 denote the probabilities associated with outcomes 1 and 2. Generalize Theorem 3.2.1 to deduce a formula for the
Repair calls for central air conditioners fall into three general categories: coolant leakage, compressor failure, and electrical malfunction. Experience has shown that the probabilities associated with the three are 0.5, 0.3, and 0.2, respectively. Suppose that a dispatcher has logged in ten
In a nuclear reactor, the fission process is controlled by inserting special rods into the radioactive core to absorb neutrons and slow down the nuclear chain reaction. When functioning properly, these rods serve as a first-line defense against a core meltdown. Suppose a reactor has ten control
A corporate board contains twelve members. The board decides to create a five-person Committee to Hide Corporation Debt. Suppose four members of the board are accountants. What is the probability that the Committee will contain two accountants and three non accountants?
One of the popular tourist attractions in Alaska is watching black bears catch salmon swimming upstream to spawn. Not all “black” bears are black, though— some are tan-colored. Suppose that six black bears and three tan-colored bears are working the rapids of a salmon stream. Over the course
A city has 4050 children under the age of ten, including 514 who have not been vaccinated for measles. Sixty-five of the city’s children are enrolled in the ABC Day Care Center. Suppose the municipal health department sends a doctor and a nurse to ABC to immunize any child who has not already
Country A inadvertently launches ten guided missiles—six armed with nuclear warheads—at Country B. In response, Country B fires seven antiballistic missiles, each of which will destroy exactly one of the incoming rockets. The antiballistic missiles have no way of detecting, though, which of the
Anne is studying for a history exam covering the French Revolution that will consist of five essay questions selected at random from a list of ten the professor has handed out to the class in advance. Not exactly a Napoleon buff, Anne would like to avoid researching all ten questions but still be
Each year a college awards five merit-based scholarships to members of the entering freshman class who have exceptional high school records. The initial pool of applicants for the upcoming academic year has been reduced to a “short list” of eight men and ten women, all of whom seem equally
Keno is a casino game in which the player has a card with the numbers 1 through 80 on it. The player selects a set of k numbers from the card, where k can range from one to fifteen. The “caller” announces twenty winning numbers, chosen at random from the eighty. The amount won depends on how
A display case contains thirty-five gems, of which ten are real diamonds and twenty-five are fake diamonds. A burglar removes four gems at random, one at a time and without replacement. What is the probability that the last gem she steals is the second real diamond in the set of four?
A bleary-eyed student awakens one morning, late for an 8:00 class, and pulls two socks out of a drawer that contains two black, six brown, and two blue socks, all randomly arranged. What is the probability that the two he draws are a matched pair?
Show directly that the set of probabilities associated with the hypergeometric distribution sum to 1.(1+μ) N = (1+μ) r (1+μ) N−r
Show that the ratio of two successive hypergeometric probability terms satisfies the following equation,for any k where both numerators are defined.
Urn I contains five red chips and four white chips; urn II contains four red and five white chips. Two chips are drawn simultaneously from urn I and placed into urn II. Then a single chip is drawn from urn II. What is the probability that the chip drawn from urn II is white?
As the owner of a chain of sporting goods stores, you have just been offered a “deal” on a shipment of one hundred robot table tennis machines. The price is right, but the prospect of picking up the merchandise at midnight from an unmarked van parked on the side of the New Jersey Turnpike is a
Suppose that r of N chips are red. Divide the chips into three groups of sizes n1, n2, and n3, where n1 + n2 + n3= N. Generalize the hypergeometric distribution to find the probability that the first group contains r1 red chips, the second group r2 red chips, and the third group r3 red chips, where
Some nomadic tribes, when faced with a life threatening, contagious disease, try to improve their chances of survival by dispersing into smaller groups. Suppose a tribe of twenty-one people, of whom four are carriers of the disease, split into three groups of seven each. What is the probability
Suppose a population contains n1 objects of one kind, n2 objects of a second kind, . . . , and nt objects of a tth kind, where n1 +n2 +· · ·+nt = N. A sample of size n is drawn at random and without replacement. Deduce an expression for the probability of drawing k1 objects of the first kind, k2
Sixteen students—five freshmen, four sophomores, four juniors, and three seniors—have applied for membership in their school’s Communications Board, a group that oversees the college’s newspaper, literary magazine, and radio show. Eight positions are open. If the selection is done at

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