Mathematical Statistics with Applications

7th Edition

ISBN: 9780495110811

Author: Dennis Wackerly, William Mendenhall, Richard L. Scheaffer

Publisher: Cengage Learning

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Textbook Question

Chapter 6.4, Problem 24E

In Exercise 6.4, we considered a random variable Y that possessed an exponential distribution with mean 4 and used the method of distribution functions to derive the density function for U = 3Y + 1. Use the method of transformations to derive the density function for U.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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Chapter 6.4, Problem 23E

Chapter 6.4, Problem 25E

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Chapter 6 Solutions

Mathematical Statistics with Applications

Ch. 6.3 - Let Y be a random variable with probability...Ch. 6.3 - Prob. 2E Ch. 6.3 - Prob. 3E Ch. 6.3 - The amount of flour used per day by a bakery is a...Ch. 6.3 - Prob. 5E Ch. 6.3 - The joint distribution of amount of pollutant...Ch. 6.3 - Suppose that Z has a standard normal distribution....Ch. 6.3 - Assume that Y has a beta distribution with...Ch. 6.3 - Prob. 9E Ch. 6.3 - The total time from arrival to completion of...

Ch. 6.3 - Suppose that two electronic components in the...Ch. 6.3 - Prob. 12E Ch. 6.3 - If Y1 and Y2 are independent exponential random...Ch. 6.3 - Prob. 14E Ch. 6.3 - Prob. 15E Ch. 6.3 - Prob. 16E Ch. 6.3 - Prob. 17E Ch. 6.3 - A member of the Pareto family of distributions...Ch. 6.3 - Prob. 19E Ch. 6.3 - Let the random variable Y possess a uniform...Ch. 6.3 - Prob. 21E Ch. 6.4 - Prob. 23E Ch. 6.4 - In Exercise 6.4, we considered a random variable Y...Ch. 6.4 - Prob. 25E Ch. 6.4 - Prob. 26E Ch. 6.4 - Prob. 27E Ch. 6.4 - Let Y have a uniform (0, 1) distribution. Show...Ch. 6.4 - Prob. 29E Ch. 6.4 - A fluctuating electric current I may be considered...Ch. 6.4 - The joint distribution for the length of life of...Ch. 6.4 - Prob. 32E Ch. 6.4 - The proportion of impurities in certain ore...Ch. 6.4 - A density function sometimes used by engineers to...Ch. 6.4 - Prob. 35E Ch. 6.4 - Refer to Exercise 6.34. Let Y1 and Y2 be...Ch. 6.5 - Let Y1, Y2,, Yn be independent and identically...Ch. 6.5 - Let Y1 and Y2 be independent random variables with...Ch. 6.5 - Prob. 39E Ch. 6.5 - Prob. 40E Ch. 6.5 - Prob. 41E Ch. 6.5 - A type of elevator has a maximum weight capacity...Ch. 6.5 - Prob. 43E Ch. 6.5 - Prob. 44E Ch. 6.5 - The manager of a construction job needs to figure...Ch. 6.5 - Suppose that Y has a gamma distribution with =...Ch. 6.5 - A random variable Y has a gamma distribution with ...Ch. 6.5 - Prob. 48E Ch. 6.5 - Let Y1 be a binomial random variable with n1...Ch. 6.5 - Let Y be a binomial random variable with n trials...Ch. 6.5 - Prob. 51E Ch. 6.5 - Prob. 52E Ch. 6.5 - Let Y1,Y2,,Yn be independent binomial random...Ch. 6.5 - Prob. 54E Ch. 6.5 - Customers arrive at a department store checkout...Ch. 6.5 - The length of time necessary to tune up a car is...Ch. 6.5 - Prob. 57E Ch. 6.5 - Prob. 58E Ch. 6.5 - Prob. 59E Ch. 6.5 - Prob. 60E Ch. 6.5 - Prob. 61E Ch. 6.5 - Prob. 62E Ch. 6.6 - In Example 6.14, Y1 and Y2 were independent...Ch. 6.6 - Refer to Exercise 6.63 and Example 6.14. Suppose...Ch. 6.6 - Prob. 65E Ch. 6.6 - Prob. 66E Ch. 6.6 - Prob. 67E Ch. 6.6 - Prob. 68E Ch. 6.6 - Prob. 71E Ch. 6 - Let Y1 and Y2 be independent and uniformly...Ch. 6 - As in Exercise 6.72, let Y1 and Y2 be independent...Ch. 6 - Let Y1, Y2,, Yn be independent, uniformly...Ch. 6 - Prob. 75SE Ch. 6 - Prob. 76SE Ch. 6 - Prob. 77SE Ch. 6 - Prob. 78SE Ch. 6 - Refer to Exercise 6.77. If Y1,Y2,,Yn are...Ch. 6 - Prob. 80SE Ch. 6 - Let Y1, Y2,, Yn be independent, exponentially...Ch. 6 - Prob. 82SE Ch. 6 - Prob. 83SE Ch. 6 - Prob. 84SE Ch. 6 - Let Y1 and Y2 be independent and uniformly...Ch. 6 - Prob. 86SE Ch. 6 - Prob. 87SE Ch. 6 - Prob. 88SE Ch. 6 - Let Y1, Y2, . . . , Yn denote a random sample from...Ch. 6 - Prob. 90SE Ch. 6 - Prob. 91SE Ch. 6 - Prob. 92SE Ch. 6 - Prob. 93SE Ch. 6 - Prob. 94SE Ch. 6 - Prob. 96SE Ch. 6 - Prob. 97SE Ch. 6 - Prob. 98SE Ch. 6 - Prob. 99SE Ch. 6 - The time until failure of an electronic device has...Ch. 6 - Prob. 101SE Ch. 6 - Prob. 103SE Ch. 6 - Prob. 104SE Ch. 6 - Prob. 105SE Ch. 6 - Prob. 106SE Ch. 6 - Prob. 107SE Ch. 6 - Prob. 108SE Ch. 6 - Prob. 109SE Ch. 6 - Prob. 110SE Ch. 6 - Prob. 111SE Ch. 6 - Prob. 112SE Ch. 6 - Prob. 113SE Ch. 6 - Prob. 114SE Ch. 6 - Prob. 115SE Ch. 6 - Prob. 116SE

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In Exercise 6.4, we considered a random variable Y that possessed an exponential distribution with mean 4 and used the method of distribution functions to derive the density function for U = 3 Y + 1. Use the method of transformations to derive the density function for U.

Solution Summary: The author calculates the density function for the random variable U by using the method of transformations.

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