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

A point is chosen at random from the interior of a circle whose equation is x2 + y2 ≤ 4. Let the random variables X and Y denote the x- and y-coordinates of the sampled point. Find f X,Y (x, y).
Find P(X < 2Y) if f X,Y (x, y) = x + y for X and Y each defined over the unit interval.
Suppose that five independent observations are drawn from the continuous pdf fT (t) =2t, 0≤t ≤1. Let X denote the number of t’s that fall in the interval 0≤t < 1/3 and let Y denote the number of t’s that fall in the interval 1/3 ≤t < 2/3. Find p X,Y (1, 2).
A point is chosen at random from the interior of a right triangle with base b and height h. What is the probability that the y value is between 0 and h/2?
Find the marginal pdf of X for the joint pdf derived in Question 3.7.5. In Question 3.7.5.an urn contains four red chips, three white chips, and two blue chips. A random sample of size 3 is drawn without replacement. Let X denote the number of white chips in the sample and Y the number of blue
The campus recruiter for an international conglomerate classifies the large number of students she interviews into three categories—the lower quarter, the middle half, and the upper quarter. If she meets six students on a given morning, what is the probability that they will be evenly divided
For each of the following joint pdfs, find fX (x) and fY (y). (a) f X,Y (x, y)= 1/2 , 0≤ x ≤2, 0≤ y ≤1 (b) f X,Y (x, y)= 3/2 y2, 0≤ x ≤2, 0≤ y ≤1 (c) f X,Y (x, y)= 2/3 (x +2y), 0≤ x ≤1, 0≤ y ≤1 (d) f X,Y (x, y)=c(x + y), 0≤ x ≤1, 0≤ y ≤1 (e) f X,Y (x, y)=4xy, 0≤ x
Let X and Y be two continuous random variables defined over the unit square. What does c equal if fX,Y (x, y)=c(x2 + y2)?
For each of the following joint pdfs, find fX (x) and fY (y). (a) f X,Y (x, y)= 1/2 , 0≤ x ≤ y ≤2 (b) f X,Y (x, y)= 1/x , 0≤ y ≤ x ≤1 (c) f X,Y (x, y)=6x, 0≤ x ≤1, 0≤ y ≤1−x
Suppose that f X,Y (x, y)=6(1− x − y) for x and y defined over the unit square, subject to the restriction that 0≤ x + y ≤1. Find the marginal pdf for X.
Find fY (y) if f X,Y (x, y) = 2eˆ’xeˆ’y for x and y defined over the shaded region pictured.
Suppose that X and Y are discrete random variables withFind pX (x) and pY (x).
A generalization of the binomial model occurs when there is a sequence of n independent trials with three outcomes, where p1 = P (outcome 1) and p2 = P (outcome 2). Let X and Y denote the number of trials (out of n) resulting in outcome 1 and outcome 2, respectively.Show that
Consider the experiment of simultaneously tossing a fair coin and rolling a fair die. Let X denote the number of heads showing on the coin and Y the number of spots showing on the die. (a) List the outcomes in S. (b) Find FX,Y (1, 2).
An urn contains twelve chips—four red, three black, and five white. A sample of size 4 is to be drawn without replacement. Let X denote the number of white chips in the sample, Y the number of red. Find FX,Y (1, 2).
For each of the following joint pdfs, find FX,Y (u, v). (a) f X,Y (x, y)= 3/2 y2, 0≤ x ≤2, 0≤ y ≤1 (b) f X,Y (x, y)= 2/3 (x +2y), 0≤ x ≤1, 0≤ y ≤1 (c) f X,Y (x, y)=4xy, 0≤ x ≤1, 0≤ y ≤1
For each of the following joint pdfs, find FX,Y (u, v). (a) f X,Y (x, y)= 1/2 , 0≤ x ≤ y ≤2 (b) f X,Y (x, y)= 1/x , 0≤ y ≤ x ≤1 (c) f X,Y (x, y)=6x, 0≤ x ≤1, 0≤ y ≤1−x
Find and graph f X,Y (x, y) if the joint cdf for random variables X and Y is FX,Y (x, y)=xy, 0
Suppose that random variables X and Y vary in accordance with the joint pdf, f X,Y (x, y)=c(x + y), 0
Find the joint pdf associated with two random variables X and Y whose joint cdf is FX,Y (x, y)=(1−e−λy)(1−e−λx ), x >0, y >0
Given that FX,Y (x, y) = k(4x2 y2 + 5xy4), 0 < x < 1, 0 < y < 1, find the corresponding pdf and use it to calculate P(0< X < 1/2 , 1/2
Prove thatP (a < X ≤b, c<Y ≤d) =F X,Y (b, d)− FX,Y (a, d) − FX,Y (b, c)+ FX,Y (a, c)
A certain brand of fluorescent bulbs will last, on the average, 1000 hours. Suppose that four of these bulbs are installed in an office. What is probability that all four are still functioning after 1050 hours? If Xi denotes the ith bulb€™s life, assume thatFor xi >0, i =1, 2, 3, 4.
A hand of six cards is dealt from a standard poker deck. Let X denote the number of aces, Y the number of kings, and Z the number of queens. (a) Write a formula for p X,Y,Z (x, y, z). (b) Find p X,Y (x, y) and p X,Z (x, z).
Calculate pX,Y (0, 1) if pX,Y,Z (x, y, z) =For x, y, z =0, 1, 2, 3 and 0‰¤ x + y +z ‰¤3.
Suppose that the random variables X, Y, and Z have the multivariate pdf fX,Y,Z (x, y, z)=(x + y)e−z For 0 < x < 1, 0 < y < 1, and z > 0. Find (a) fX,Y (x, y), (b) fY,Z (y, z), and (c) fZ (z).
The four random variables W, X, Y, and Z have the multivariate pdf fW,X,Y,Z(w, x, y, z)=16wxyz For 0
Let fX,Y (x, y) = λ2e−λ(x+y), 0 ≤ x, 0 ≤ y. Show that X and Y are independent. What are the marginal pdfs in this case?
Find c if fX,Y (x, y)= cxy for X and Y defined over the triangle whose vertices are the points (0, 0), (0, 1), and (1, 1).
Suppose that each of two urns has four chips, numbered 1 through 4. A chip is drawn from the first urn and bears the number X. That chip is added to the second urn. A chip is then drawn from the second urn. Call its number Y. (a) Find pX,Y (x, y). (b) Show that pX (k) = pY (k) = 1/4, k =1, 2, 3,
Let X and Y be random variables with joint pdf fX,Y (x, y)=k, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ x + y ≤ 1 Give a geometric argument to show that X and Y are not independent.
Are the random variables X and Y independent if fX,Y (x, y) = 2/3 (x + 2y), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1?
Find the joint cdf of the independent random variables X and Y, where fX (x) = x/2, 0 ≤ x ≤ 2, and fY (y) = 2y, 0 ≤ y ≤ 1.
Suppose fX,Y (x, y)= xye−(x+y), x >0, y >0. Prove for any real numbers a, b, c, and d that P(a < X < b, c < Y < d) = P(a < X < b) · P(c < Y < d) thereby establishing the independence of X and Y.
Given the joint pdf fX,Y (x, y) = 2x + y −2xy, 0 < x <1, 0< y <1, find numbers a, b, c, and d such thatP(a < X < b, c < Y < d) = P(a < X < b) · P(c < Y < d)thus demonstrating that X and Y are not independent.
Prove that if X and Y are two independent random variables, then U = g(X) and V = h(Y) are also independent.
If two random variables X and Y are defined over a region in the XY-plane that is not a rectangle (possibly infinite) with sides parallel to the coordinate axes, can X and Y be independent?
An urn contains four red chips, three white chips, and two blue chips. A random sample of size 3 is drawn without replacement. Let X denote the number of white chips in the sample and Y the number of blue chips. Write a formula for the joint pdf of X and Y.
Suppose that X1, X2, X3, and X4 are independent random variables, each with pdf fXi (xi) = 4x3i, 0 ≤ xi ≤ 1. Find(a) P(X1 < ½).(b) P(exactly one Xi < 1/2)(c) fX1, X2, X3, X4 (x1, x2, x3, x4).(d) FX2, X3 (x2, x3).
A random sample of size n = 2k is taken from a uniform pdf defined over the unit interval. Calculate P P(X1 < 1/2, X2 > 1/2, X3 < 1/2, X4 > 1/2, …, X2k > 1/2)
Four cards are drawn from a standard poker deck. Let X be the number of kings drawn and Y the number of queens. Find p X,Y (x, y).
An advisor looks over the schedules of his fifty students to see how many math and science courses each has registered for in the coming semester. He summarizes his results in a table. What is the probability that a student selected at random will have signed up for more math courses than science
Consider the experiment of tossing a fair coin three times. Let X denote the number of heads on the last flip, and let Y denote the total number of heads on the three flips. Find p X,Y (x, y).
Suppose that two fair dice are tossed one time. Let X denote the number of 2’s that appear, and Y the number of 3’s. Write the matrix giving the joint probability density function for X and Y. Suppose a third random variable, Z, is defined, where Z = X +Y. Use p X,Y (x, y) to find pZ (z).
Let X and Y be two independent random variables. Given the marginal pdfs shown below, find the pdf of X + Y. In each case, check to see if X + Y belongs to the same family of pdfs as do X and Y. (a) pX(k) = e−λ λk/k! and pY(k) = e−μ μk/k! , k = 0, 1, 2, . . . (b) pX(k) = pY(k) = (1 −
Let X and Y be two independent random variables. Given the marginal pdfs indicated below, find the cdf of Y/X. (a) fX(x) = 1, 0 ≤ x ≤ 1, and fY(y) = 1, 0 ≤ y ≤ 1 (b) fX(x) = 2x, 0 ≤ x ≤ 1, and fY(y) = 2y, 0 ≤ y ≤ 1
Suppose that X and Y are two independent random variables, where fX(x) = xe−x, x ≥ 0, and fY(y) = e−y, y ≥ 0. Find the pdf of Y/X.
Suppose fX(x) = xe−x, x ≥ 0, and fY(y) = e−y, y ≥ 0, where X and Y are independent. Find the pdf of X + Y.
Let X and Y be two independent random variables, whose marginal pdfs are given below. Find the pdf of X + Y. (Consider two cases, 0 ≤ w < 1 and 1 ≤ w ≤ 2.) fX(x) = 1, 0 ≤ x ≤ 1, and fY(y) = 1, 0 ≤ y ≤ 1
If a random variable V is independent of two independent random variables X and Y, prove that V is independent of X + Y.
Let Y be a continuous nonnegative random variable. Show that W = Y2 has pdf fW(w) = 1/2√√w fY(√w).
Let Y be a uniform random variable over the interval [0, 1]. Find the pdf of W = Y2.
Let Y be a random variable with fY(y) = 6y(1 − y), 0 ≤ y ≤ 1. Find the pdf of W = Y2.
Suppose the velocity of a gas molecule of mass m is a random variable with pdf fY(y) = ay2e−by2, y ≥ 0, where a and b are positive constants depending on the gas. Find the pdf of the kinetic energy, W = (m/2)Y2, of such a molecule.
Given that X and Y are independent random variables, find the pdf of XY for the following two sets of marginal pdfs: (a) fX(x) = 1, 0 ≤ x ≤ 1, and fY(y) = 1, 0 ≤ y ≤ 1 (b) fX(x) = 2x, 0 ≤ x ≤ 1, and fY(y) = 2y, 0 ≤ y ≤ 1
Suppose that r chips are drawn with replacement from an urn containing n chips, numbered 1 through n. Let V denote the sum of the numbers drawn. Find E(V).
Suppose that X and Y are both uniformly distributed over the interval [0, 1]. Calculate the expected value of the square of the distance of the random point (X, Y) from the origin; that is, find E(X2 + Y2).
Suppose X represents a point picked at random from the interval [0, 1] on the x-axis, and Y is a point picked at random from the interval [0, 1] on the y-axis. Assume that X and Y are independent. What is the expected value of the area of the triangle formed by the points (X, 0), (0, Y), and (0, 0)?
Suppose Y1, Y2, . . . , Yn is a random sample from the uniform pdf over [0, 1]. The geometric mean of the numbers is the random variable n√Y1Y2 · · · · · Yn. Compare the expected value of the geometric mean to that of the arithmetic mean .
Suppose that two dice are thrown. Let X be the number showing on the first die and let Y be the larger of the two numbers showing. Find Cov(X, Y).
Show that Cov(aX + b, cY + d) = acCov(X, Y) for any constants a, b, c, and d.
Let U be a random variable uniformly distributed over [0, 2π]. Define X = cosU and Y = sin U. Show that X and Y are dependent but that Cov(X, Y) = 0.
Let X and Y be random variables withShow that Cov(X, Y) = 0 but that X and Y are dependent.
Suppose that fX,Y(x, y) = λ2e−λ(x + y), 0 ≤ x, 0 ≤ y. Find Var(X + Y).
Suppose that fX,Y(x, y) = 2/3 (x + 2y), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. Find Var(X + Y).
For the uniform pdf defined over [0, 1], find the variance of the geometric mean when n = 2
Suppose that fX,Y(x, y) = λ2e−λ(x+y), 0 ≤ x, 0 ≤ y. Find E(X + Y).
Let X be a binomial random variable based on n trials and a success probability of px; let Y be an independent binomial random variable based on m trials and a success probability of pY. Find E(W) and Var(W), where W = 4X + 6Y.
Let the Poisson random variable U be the number of calls for technical assistance received by a computer company during the firm's nine normal workday hours. Suppose the average number of calls per hour is 7.0 and that each call costs the company $50. Let V be a Poisson random variable representing
A mason is contracted to build a patio retaining wall. Plans call for the base of the wall to be a row of fifty 10-inch bricks, each separated by 1/2 -inch-thick mortar. Suppose that the bricks used are randomly chosen from a population of bricks whose mean length is 10 inches and whose standard
An electric circuit has six resistors wired in series, each nominally being five ohms. What is the maximum standard deviation that can be allowed in the manufacture of these resistors if the combined circuit resistance is to have a standard deviation no greater than 0.4 ohm?
A gambler plays n hands of poker. If he wins the kth hand, he collects k dollars; if he loses the kth hand, he collects nothing. Let T denote his total winnings in n hands. Assuming that his chances of winning each hand are constant and independent of his success or failure at any other hand, find
Suppose that fX,Y(x, y) = 2 3 (x + 2y), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 [recall Question 3.7.19(c)]. Find E(X + Y).
Marksmanship competition at a certain level requires each contestant to take ten shots with each of two different handguns. Final scores are computed by taking a weighted average of 4 times the number of bull's-eyes made with the first gun plus 6 times the number gotten with the second. If Cathie
Suppose that Xi is a random variable for which E(Xi) = Î¼, i = 1, 2, . . . , n. Under what conditions will the following be true?
Suppose that the daily closing price of stock goes up an eighth of a point with probability p and down an eighth of a point with probability q, where p > q. After n days how much gain can we expect the stock to have achieved? Assume that the daily price fluctuations are independent events.
An urn contains r red balls and w white balls. A sample of n balls is drawn in order and without replacement. Let Xi be 1 if the ith draw is red and 0 otherwise, i =1, 2, . . . , n. (a) Show that E(Xi) = E(X1), i = 2, 3, . . . , n. (b) Use the corollary to Theorem 3.9.2 to show that the expected
Suppose two fair dice are tossed. Find the expected value of the product of the faces showing.
Find E(R) for a two-resistor circuit similar to the one described in Example 3.9.2, where fX,Y(x, y) = k(x + y), 10 ≤ x ≤ 20, 10 ≤ y ≤ 20.
If a typist averages one misspelling in every 3250 words, what are the chances that a 6000-word report is free of all such errors? Answer the question two ways- first, by using an exact binomial analysis, and second, by using a Poisson approximation. Does the similarity (or dissimilarity) of the
During the latter part of the nineteenth century, Prussian officials gathered information relating to the hazards that horses posed to cavalry soldiers. A total of ten cavalry corps were monitored over a period of twenty years. Recorded for each year and each corps was X, the annual number of
A random sample of 356 seniors enrolled at the University of West Florida was categorized according to X, the number of times they had changed majors (110). Based on the summary of that information shown in the following table, would you conclude that X can be treated as a Poisson random
Midwestern Skies books ten commuter flights each week. Passenger totals are much the same from week to week, as are the numbers of pieces of luggage that are checked. Listed in the following table are the numbers of bags that were lost during each of the first forty weeks in 2009. Do these figures
In 1893, New Zealand became the first country to permit women to vote. Scattered over the ensuing 113 years, various countries joined the movement to grant this right to women. The table below (121) shows how many countries took this step in a given year. Do these data seem to follow a Poisson
The following are the daily numbers of death notices for women over the age of eighty that appeared in the London Times over a three-year period (74).(a) Does the Poisson pdf provide a good description of the variability pattern evident in these data?(b) If your answer to part (a) is "no," which of
A certain species of European mite is capable of damaging the bark on orange trees. The following are the results of inspections done on one hundred saplings chosen at random from a large orchard. The measurement recorded, X, is the number of mite infestations found on the trunk of each tree. Is it
A tool and die press that stamps out cams used in small gasoline engines tends to break down once every five hours. The machine can be repaired and put back on line quickly, but each such incident costs $50. What is the probability that maintenance expenses for the press will be no more than $100
In a new fiber-optic communication system, transmission errors occur at the rate of 1.5 per ten seconds. What is the probability that more than two errors will occur during the next half-minute?
Assume that the number of hits, X, that a baseball team makes in a nine-inning game has a Poisson distribution. If the probability that a team makes zero hits is 1 3 , what are their chances of getting two or more hits?
Flaws in metal sheeting produced by a high temperature roller occur at the rate of one per ten square feet. What is the probability that three or more flaws will appear in a five-by-eight-foot panel?
A medical study recently documented that 905 mistakes were made among the 289,411 prescriptions written during one year at a large metropolitan teaching hospital. Suppose a patient is admitted with a condition serious enough to warrant 10 different prescriptions. Approximate the probability that at
Suppose a radioactive source is metered for two hours, during which time the total number of alpha particles counted is 482. What is the probability that exactly three particles will be counted in the next two minutes? Answer the question two ways-first, by defining X to be the number of particles
Suppose that on-the-job injuries in a textile mill occur at the rate of 0.1 per day.(a) What is the probability that two accidents will occur during the next (five-day) workweek?(b) Is the probability that four accidents will occur over the next two workweeks the square of your answer to part (a)?
Find P(X = 4) if the random variable X has a Poisson distribution such that P(X = 1) = P(X = 2).
Let X be a Poisson random variable with parameter λ. Show that the probability that X is even is 1/2 (1 + e−2λ).
Let X and Y be independent Poisson random variables with parameters λ and μ, respectively. Example 3.12.10 established that X + Y is also Poisson with parameter λ + μ. Prove that same result using Theorem 3.8.3.
If X1 is a Poisson random variable for which E(X1) = λ and if the conditional pdf of X2 given that X1 = X1 is binomial with parameters X1 and p, show that the marginal pdf of X2 is Poisson with E(X2) = λp.
Suppose that commercial airplane crashes in a certain country occur at the rate of 2.5 per year. (a) Is it reasonable to assume that such crashes are Poisson events? Explain. (b) What is the probability that four or more crashes will occur next year? (c) What is the probability that the next two
Records show that deaths occur at the rate of 0.1 per day among patients residing in a large nursing home. If someone dies today, what are the chances that a week or more will elapse before another death occurs?
Suppose that Y1 and Y2 are independent exponential random variables, each having pdf fY(y) =λe−λy , y > 0. If Y = Y1 + Y2, it can be shown that fY1 + Y2 (y) =λ2ye−λy, y > 0 Recall Case Study 4.2.4. What is the probability that the next three eruptions of Mauna Loa will be less than forty

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