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Questions and Answers of
Statistics
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
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
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
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
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) = 2exey 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
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
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.
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,
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
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
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
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
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
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,
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
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.
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)
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 ≤
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) =
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
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
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
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.
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
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)
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
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
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
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
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
Suppose that Xi is a random variable for which E(Xi) = μ, 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
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
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
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
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
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
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)
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
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.
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
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
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
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
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
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
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
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
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
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
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