6th Edition

ISBN: 9780134114262

Author: LARSEN

Publisher: PEARSON

Concept explainers

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear re…

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.

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Textbook Question

Chapter 3.12, Problem 10Q

Find E ( Y 4 ) if Y is an exponential random variable with f Y ( y ) = λ e − λ y , y > 0 .

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Chapter 3.12, Problem 9Q

Chapter 3.12, Problem 11Q

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Chapter 3 Solutions

INTRO.TO MATHEMATICAL STAT...-STD.SOLN.

Ch. 3.2 - An investment analyst has tracked a certain...Ch. 3.2 - In a nuclear reactor, the fission process is...Ch. 3.2 - In 2009 a donor who insisted on anonymity gave...Ch. 3.2 - An entrepreneur owns six corporations, each with...Ch. 3.2 - The probability is 0.10 that ball bearings in a...Ch. 3.2 - Suppose that since the early 1950s some...Ch. 3.2 - Doomsday Airlines (Come Take the Flight of Your...Ch. 3.2 - Two lighting systems are being proposed for an...Ch. 3.2 - The great English diarist Samuel Pepys asked his...Ch. 3.2 - The gunner on a small assault boat fires six...

Ch. 3.2 - If a family has four children, is it more likely...Ch. 3.2 - Experience has shown that only 13 of all patients...Ch. 3.2 - Transportation to school for a rural countys...Ch. 3.2 - The captain of a Navy gunboat orders a volley of...Ch. 3.2 - A computer has generated seven random numbers over...Ch. 3.2 - Listed in the following table is the length...Ch. 3.2 - Redo Example 3.2.4 assuming n=12 and p=0.3.Ch. 3.2 - Prob. 18Q Ch. 3.2 - Prob. 19Q Ch. 3.2 - A corporate board contains twelve members. The...Ch. 3.2 - One of the popular tourist attractions in Alaska...Ch. 3.2 - A city has 4050 children under the age of ten,...Ch. 3.2 - Country A inadvertently launches ten guided...Ch. 3.2 - Anne is studying for a history exam covering the...Ch. 3.2 - Each year a college awards five merit-based...Ch. 3.2 - Keno is a casino game in which the player has a...Ch. 3.2 - A display case contains thirty-five gems, of which...Ch. 3.2 - Consider an urn with r red balls and w white...Ch. 3.2 - Prob. 29Q Ch. 3.2 - Prob. 30Q Ch. 3.2 - Prob. 31Q Ch. 3.2 - Prob. 32Q Ch. 3.2 - Prob. 33Q Ch. 3.2 - Some nomadic tribes, when faced with a...Ch. 3.2 - Suppose a population contains n1 objects of one...Ch. 3.2 - Prob. 36Q Ch. 3.3 - Prob. 1Q Ch. 3.3 - Repeat Question 3.3.1 for the case where the two...Ch. 3.3 - Suppose a fair die is tossed three times. Let X be...Ch. 3.3 - Suppose a fair die is tossed three times. Let X be...Ch. 3.3 - A fair coin is tossed three times. Let X be the...Ch. 3.3 - Suppose die one has spots 1, 2, 2, 3, 3, 4 and die...Ch. 3.3 - Suppose a particle moves along the x-axis...Ch. 3.3 - How would the pdf asked for in Question 3.3.7 be...Ch. 3.3 - Suppose that five people, including you and a...Ch. 3.3 - Prob. 10Q Ch. 3.3 - Prob. 11Q Ch. 3.3 - Prob. 12Q Ch. 3.3 - A fair die is rolled four times. Let the random...Ch. 3.3 - At the points x=0,1,...,6, the cdf for the...Ch. 3.3 - Find the pdf for the infinite-valued discrete...Ch. 3.3 - Recall the game of Fantasy Five from Example...Ch. 3.4 - Suppose fY(y)=4y3,0y1. Find P(0Y12).Ch. 3.4 - For the random variable Y with pdf...Ch. 3.4 - Let fY(y)=23y2,1y1. Find P(|Y12|14). Draw a graph...Ch. 3.4 - For persons infected with a certain form of...Ch. 3.4 - For a high-risk driver, the time in days between...Ch. 3.4 - Let n be a positive integer. Show that...Ch. 3.4 - Find the cdf for the random variable Y given in...Ch. 3.4 - If Y is an exponential random variable,...Ch. 3.4 - If the pdf for Y is fY(y)={0,|y|11|y|,|y|1 find...Ch. 3.4 - Prob. 10Q Ch. 3.4 - Prob. 11Q Ch. 3.4 - Prob. 12Q Ch. 3.4 - Prob. 13Q Ch. 3.4 - Prob. 14Q Ch. 3.4 - The logistic curve F(y)=11+ey,y, can represent a...Ch. 3.4 - Prob. 16Q Ch. 3.4 - Prob. 17Q Ch. 3.4 - Let Y be a random variable denoting the age at...Ch. 3.5 - Recall the game of Keno described in Question...Ch. 3.5 - The roulette wheels in Monte Carlo typically have...Ch. 3.5 - The pdf describing the daily profit, X, earned by...Ch. 3.5 - In the game of redball, two drawings are made...Ch. 3.5 - Suppose a life insurance company sells a $50,000,...Ch. 3.5 - A manufacturer has one hundred memory chips in...Ch. 3.5 - Records show that 642 new students have just...Ch. 3.5 - Prob. 8Q Ch. 3.5 - Recall Question 3.4.4, where the length of time Y...Ch. 3.5 - Let the random variable Y have the uniform...Ch. 3.5 - Show that the expected value associated with the...Ch. 3.5 - Show that fY(y)=1y2,y1 is a valid pdf but that Y...Ch. 3.5 - Based on recent experience, ten-year-old passenger...Ch. 3.5 - Prob. 14Q Ch. 3.5 - A city has 74,806 registered automobiles. Each is...Ch. 3.5 - Regulators have found that twenty-three of the...Ch. 3.5 - An urn contains four chips numbered 1 through 4....Ch. 3.5 - A fair coin is tossed three times. Let the random...Ch. 3.5 - Prob. 19Q Ch. 3.5 - For the St. Petersburg problem (Example 3.5.5),...Ch. 3.5 - Prob. 21Q Ch. 3.5 - Prob. 22Q Ch. 3.5 - Suppose that two evenly matched teams are playing...Ch. 3.5 - An urn contains one white chip and one black chip....Ch. 3.5 - Prob. 25Q Ch. 3.5 - Prob. 26Q Ch. 3.5 - Find the median for each of the following pdfs:...Ch. 3.5 - Suppose X is a binomial random variable with n=10...Ch. 3.5 - A typical days production of a certain electronic...Ch. 3.5 - Let Y have probability density function...Ch. 3.5 - Prob. 31Q Ch. 3.5 - A box is to be constructed so that its height is...Ch. 3.5 - Prob. 33Q Ch. 3.5 - If Y has probability density function fY(y)=2y,0y1...Ch. 3.5 - Prob. 35Q Ch. 3.5 - Prob. 36Q Ch. 3.6 - Find Var(X) for the urn problem of Example 3.6.1...Ch. 3.6 - Find the variance of Y if...Ch. 3.6 - Ten equally qualified applicants, six men and four...Ch. 3.6 - A certain hospitalization policy pays a cash...Ch. 3.6 - Use Theorem 3.6.1 to find the variance of the...Ch. 3.6 - If fY(y)=2yk2,0yk for what value of k does...Ch. 3.6 - Calculate the standard deviation, , for the random...Ch. 3.6 - Consider the pdf defined by fY(y)=2y3,y1 Show that...Ch. 3.6 - Frankie and Johnny play the following game....Ch. 3.6 - Let Y be a random variable whose pdf is given by...Ch. 3.6 - Suppose that Y is an exponential random variable,...Ch. 3.6 - Suppose that Y is an exponential random variable...Ch. 3.6 - Let X be a random variable with finite mean ....Ch. 3.6 - Suppose the charge for repairing an automobile...Ch. 3.6 - If Y denotes a temperature recorded in degrees...Ch. 3.6 - Prob. 16Q Ch. 3.6 - Suppose U is a uniform random variable over [0,1]....Ch. 3.6 - Recovering small quantities of calcium in the...Ch. 3.6 - Let Y be a uniform random variable defined over...Ch. 3.6 - Find the coefficient of skewness for an...Ch. 3.6 - Calculate the coefficient of kurtosis for a...Ch. 3.6 - Suppose that W is a random variable for which...Ch. 3.6 - If Y=aX+b,a0, show that Y has the same...Ch. 3.6 - Let Y be the random variable of Question 3.4.6,...Ch. 3.6 - Prob. 25Q Ch. 3.7 - Prob. 1Q Ch. 3.7 - Prob. 2Q Ch. 3.7 - Prob. 3Q Ch. 3.7 - Find c if fX,Y(x,y)=cxy for X and Y defined over...Ch. 3.7 - Prob. 5Q Ch. 3.7 - Four cards are drawn from a standard poker deck....Ch. 3.7 - An advisor looks over the schedules of his fifty...Ch. 3.7 - Consider the experiment of tossing a fair coin...Ch. 3.7 - Suppose that two fair dice are tossed one time....Ch. 3.7 - Let X be the time in days between a car accident...Ch. 3.7 - Let X and Y have the joint pdf...Ch. 3.7 - A point is chosen at random from the interior of a...Ch. 3.7 - Find P(X2Y) if fX,Y(x,y)=x+y for X and Y each...Ch. 3.7 - Prob. 14Q Ch. 3.7 - A point is chosen at random from the interior of a...Ch. 3.7 - Prob. 16Q Ch. 3.7 - Find the marginal pdfs of X and Y for the joint...Ch. 3.7 - Prob. 18Q Ch. 3.7 - For each of the following joint pdfs, find fX(x)...Ch. 3.7 - For each of the following joint pdfs, find fX(x)...Ch. 3.7 - Prob. 21Q Ch. 3.7 - Prob. 22Q Ch. 3.7 - Prob. 23Q Ch. 3.7 - Prob. 24Q Ch. 3.7 - Consider the experiment of simultaneously tossing...Ch. 3.7 - Prob. 26Q Ch. 3.7 - For each of the following joint pdfs, find...Ch. 3.7 - Prob. 28Q Ch. 3.7 - Prob. 29Q Ch. 3.7 - Prob. 30Q Ch. 3.7 - Given that FX,Y(x,y)=k(4x2y2+5xy4),0x1,0y1, find...Ch. 3.7 - Prob. 32Q Ch. 3.7 - Prob. 33Q Ch. 3.7 - Prob. 34Q Ch. 3.7 - Prob. 35Q Ch. 3.7 - Prob. 36Q Ch. 3.7 - Prob. 37Q Ch. 3.7 - Prob. 38Q Ch. 3.7 - Prob. 39Q Ch. 3.7 - Suppose that each of two urns has four chips,...Ch. 3.7 - Let X and Y be random variables with joint pdf...Ch. 3.7 - Are the random variables X and Y independent if...Ch. 3.7 - Prob. 43Q Ch. 3.7 - Find the joint cdf of the independent random...Ch. 3.7 - Prob. 45Q Ch. 3.7 - Prob. 46Q Ch. 3.7 - Prob. 47Q Ch. 3.7 - Prob. 48Q Ch. 3.7 - Prob. 49Q Ch. 3.7 - Prob. 50Q Ch. 3.7 - Suppose that X1,X2,X3, and X4 are independent...Ch. 3.7 - Prob. 52Q Ch. 3.8 - Prob. 1Q Ch. 3.8 - Prob. 2Q Ch. 3.8 - Prob. 3Q Ch. 3.8 - Prob. 4Q Ch. 3.8 - Prob. 5Q Ch. 3.8 - Prob. 6Q Ch. 3.8 - Prob. 7Q Ch. 3.8 - Prob. 8Q Ch. 3.8 - Prob. 9Q Ch. 3.8 - Prob. 10Q Ch. 3.8 - Prob. 11Q Ch. 3.8 - Prob. 12Q Ch. 3.8 - Prob. 13Q Ch. 3.9 - Prob. 1Q Ch. 3.9 - Prob. 2Q Ch. 3.9 - Suppose that fX,Y(x,y)=23(x+2y),0x1,0y1 [recall...Ch. 3.9 - Marksmanship competition at a certain level...Ch. 3.9 - Suppose that Xi is a random variable for which...Ch. 3.9 - Prob. 6Q Ch. 3.9 - Prob. 7Q Ch. 3.9 - Suppose two fair dice are tossed. Find the...Ch. 3.9 - Prob. 9Q Ch. 3.9 - Suppose that X and Y are both uniformly...Ch. 3.9 - Prob. 11Q Ch. 3.9 - Prob. 12Q Ch. 3.9 - Prob. 13Q Ch. 3.9 - Prob. 14Q Ch. 3.9 - Prob. 15Q Ch. 3.9 - Let X and Y be random variables with...Ch. 3.9 - Suppose that fX,Y(x,y)=2e(x+y),0x,0y. Find...Ch. 3.9 - Prob. 18Q Ch. 3.9 - Prob. 19Q Ch. 3.9 - Let X be a binomial random variable based on n...Ch. 3.9 - Prob. 21Q Ch. 3.9 - Prob. 22Q Ch. 3.9 - Prob. 23Q Ch. 3.9 - A gambler plays n hands of poker. If he wins the...Ch. 3.10 - Suppose the length of time, in minutes, that you...Ch. 3.10 - A random sample of size n=6 is taken from the pdf...Ch. 3.10 - What is the probability that the larger of two...Ch. 3.10 - Prob. 4Q Ch. 3.10 - Prob. 5Q Ch. 3.10 - Let Y1,Y2,...,Yn be a random sample from the...Ch. 3.10 - Calculate P(0.6Y40.7) if a random sample of size 6...Ch. 3.10 - A random sample of size n=5 is drawn from the pdf...Ch. 3.10 - Prob. 9Q Ch. 3.10 - Suppose that n observations are chosen at random...Ch. 3.10 - In a certain large metropolitan area, the...Ch. 3.10 - Consider a system containing n components, where...Ch. 3.10 - Prob. 13Q Ch. 3.10 - Prob. 14Q Ch. 3.10 - Prob. 15Q Ch. 3.10 - Suppose a device has three independent components,...Ch. 3.11 - Prob. 1Q Ch. 3.11 - Suppose a die is rolled six times. Let X be the...Ch. 3.11 - Prob. 3Q Ch. 3.11 - Five cards are dealt from a standard poker deck....Ch. 3.11 - Given that two discrete random variables X and Y...Ch. 3.11 - Prob. 6Q Ch. 3.11 - Suppose X, Y, and Z have a trivariate distribution...Ch. 3.11 - Prob. 8Q Ch. 3.11 - Let X and Y be independent Poisson random...Ch. 3.11 - Prob. 10Q Ch. 3.11 - Prob. 11Q Ch. 3.11 - Prob. 12Q Ch. 3.11 - Prob. 13Q Ch. 3.11 - Prob. 14Q Ch. 3.11 - Prob. 15Q Ch. 3.11 - Prob. 16Q Ch. 3.11 - Prob. 17Q Ch. 3.11 - Prob. 18Q Ch. 3.11 - Prob. 19Q Ch. 3.11 - Prob. 20Q Ch. 3.11 - For continuous random variables X and Y, prove...Ch. 3.12 - Let X be a random variable with pdf pX(k)=1/n, for...Ch. 3.12 - Two chips are drawn at random and without...Ch. 3.12 - Prob. 3Q Ch. 3.12 - Find the moment-generating function for the...Ch. 3.12 - Which pdfs would have the following...Ch. 3.12 - Prob. 6Q Ch. 3.12 - The random variable X has a Poisson distribution...Ch. 3.12 - Prob. 8Q Ch. 3.12 - Prob. 9Q Ch. 3.12 - Find E(Y4) if Y is an exponential random variable...Ch. 3.12 - Prob. 11Q Ch. 3.12 - Prob. 12Q Ch. 3.12 - Prob. 13Q Ch. 3.12 - Prob. 14Q Ch. 3.12 - Prob. 15Q Ch. 3.12 - Find the variance of Y if MY(t)=e2t/(1t2).Ch. 3.12 - Prob. 17Q Ch. 3.12 - Let Y1,Y2, and Y3 be independent random variables,...Ch. 3.12 - Use Theorems 3.12.2 and 3.12.3 to determine which...Ch. 3.12 - Calculate P(X2) if MX(t)=(14+34et)5.Ch. 3.12 - Suppose that Y1,Y2,...,Yn is a random sample of...Ch. 3.12 - Suppose the moment-generating function for a...Ch. 3.12 - Suppose that X is a Poisson random variable, where...Ch. 3.12 - Prob. 24Q

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Find E ( Y 4 ) if Y is an exponential random variable with f Y ( y ) = λ e − λ y , y &gt; 0 .

Solution Summary: The author explains that the moment-generating function is E(Y4)=24lambda 4,y>0.

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Chapter 3 Solutions

Find E ( Y 4 ) if Y is an exponential random variable with f Y ( y ) = λ e − λ y , y > 0 .