An Introduction to Mathematical Statistics and Its Applications (6th Edition)

6th Edition

ISBN: 9780134114217

Author: Richard J. Larsen, Morris L. Marx

Publisher: PEARSON

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Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

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Textbook Question

Chapter 3.12, Problem 1Q

Let X be a random variable with pdf p X ( k ) = 1 / n , for k = 0 , 1 , 2 , ... , n − 1 and 0 otherwise. Show that M X ( t ) = 1 − e n t n ( 1 − e t ) .

Definition Definition Probability of occurrence of a continuous random variable within a specified range. When the value of a random variable, Y, is evaluated at a point Y=y, then the probability distribution function gives the probability that Y will take a value less than or equal to y. The probability distribution function formula for random Variable Y following the normal distribution is: F(y) = P (Y ≤ y) The value of probability distribution function for random variable lies between 0 and 1.

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Chapter 3.11, Problem 21Q

Chapter 3.12, Problem 2Q

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Chapter 3 Solutions

An Introduction to Mathematical Statistics and Its Applications (6th Edition)

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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