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 5.4, Problem 43E

Let Y₁ and Y₂ have joint density function f(y₁, y₂) and marginal densities f₁(y₁) and f₂(y₂), respectively. Show that Y₁ and Y₂ are independent if and only if f(y₁|y₂) = f₁(y₁) for all values of y₁ and for all y₂ such that f₂(y₂) > 0. A completely analogous argument establishes that Y₁ and Y₂ are independent if and only if f(y₂|y₁) = f₂(y₂) for all values of y₂ and for all y₁ such that f₁(y₁) > 0.

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Chapter 5.3, Problem 42E

Chapter 5.4, Problem 44E

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Chapter 5 Solutions

Mathematical Statistics with Applications

Ch. 5.2 - Contracts for two construction jobs are randomly...Ch. 5.2 - Three balanced coins are tossed independently. One...Ch. 5.2 - Of nine executives in a business firm, four are...Ch. 5.2 - Given here is the joint probability function...Ch. 5.2 - Refer to Example 5.4. The joint density of Y1, the...Ch. 5.2 - Prob. 6E Ch. 5.2 - Let Y1 and Y2 have joint density function...Ch. 5.2 - Prob. 8E Ch. 5.2 - Prob. 9E Ch. 5.2 - An environmental engineer measures the amount (by...

Ch. 5.2 - Suppose that Y1 and Y2 are uniformly distributed...Ch. 5.2 - Prob. 12E Ch. 5.2 - Prob. 13E Ch. 5.2 - Prob. 14E Ch. 5.2 - The management at a fast-food outlet is interested...Ch. 5.2 - Let Y1 and Y2 denote the proportions of time (out...Ch. 5.2 - Let (Y1, Y2) denote the coordinates of a point...Ch. 5.2 - Prob. 18E Ch. 5.3 - In Exercise 5.1, we determined that the joint...Ch. 5.3 - Refer to Exercise 5.2. a Derive the marginal...Ch. 5.3 - In Exercise 5.3, we determined that the joint...Ch. 5.3 - In Exercise 5.4, you were given the following...Ch. 5.3 - In Example 5.4 and Exercise 5.5, we considered the...Ch. 5.3 - Prob. 24E Ch. 5.3 - Prob. 25E Ch. 5.3 - Prob. 26E Ch. 5.3 - Prob. 27E Ch. 5.3 - In Exercise 5.10, we proved that...Ch. 5.3 - Prob. 29E Ch. 5.3 - In Exercise 5.12, we were given the following...Ch. 5.3 - In Exercise 5.13, the joint density function of Y1...Ch. 5.3 - Prob. 32E Ch. 5.3 - Suppose that Y1 is the total time between a...Ch. 5.3 - Prob. 34E Ch. 5.3 - Prob. 35E Ch. 5.3 - Prob. 36E Ch. 5.3 - Prob. 37E Ch. 5.3 - Let Y1 denote the weight (in tons) of a bulk item...Ch. 5.3 - Prob. 39E Ch. 5.3 - Prob. 40E Ch. 5.3 - Prob. 41E Ch. 5.3 - Prob. 42E Ch. 5.4 - Let Y1 and Y2 have joint density function f(y1,...Ch. 5.4 - Prob. 44E Ch. 5.4 - Prob. 45E Ch. 5.4 - Prob. 46E Ch. 5.4 - In Exercise 5.3, we determined that the joint...Ch. 5.4 - In Exercise 5.4, you were given the following...Ch. 5.4 - In Example 5.4 and Exercise 5.5, we considered the...Ch. 5.4 - Prob. 50E Ch. 5.4 - Prob. 51E Ch. 5.4 - Prob. 52E Ch. 5.4 - Prob. 53E Ch. 5.4 - Prob. 54E Ch. 5.4 - Prob. 55E Ch. 5.4 - In Exercise 5.12, we were given the following...Ch. 5.4 - Prob. 57E Ch. 5.4 - Suppose that the random variables Y1 and Y2 have...Ch. 5.4 - If Y1 is the total time between a customers...Ch. 5.4 - Prob. 60E Ch. 5.4 - Prob. 61E Ch. 5.4 - Prob. 62E Ch. 5.4 - Let Y1 and Y2 be independent exponentially...Ch. 5.4 - Prob. 64E Ch. 5.4 - Prob. 65E Ch. 5.4 - Let F1(y1) and F2(y2) be two distribution...Ch. 5.4 - Prob. 67E Ch. 5.4 - Prob. 68E Ch. 5.4 - The length of life Y for fuses of a certain type...Ch. 5.4 - A bus arrives at a bus stop at a uniformly...Ch. 5.4 - Prob. 71E Ch. 5.6 - In Exercise 5.1, we determined that the joint...Ch. 5.6 - Prob. 73E Ch. 5.6 - Refer to Exercises 5.6, 5.24, and 5.50. Suppose...Ch. 5.6 - Prob. 75E Ch. 5.6 - Prob. 76E Ch. 5.6 - Prob. 77E Ch. 5.6 - Prob. 78E Ch. 5.6 - Suppose that, as in Exercise 5.11, Y1 and Y2 are...Ch. 5.6 - In Exercise 5.16, Y1 and Y2 denoted the...Ch. 5.6 - In Exercise 5.18, Y1 and Y2 denoted the lengths of...Ch. 5.6 - In Exercise 5.38, we determined that the joint...Ch. 5.6 - Prob. 83E Ch. 5.6 - In Exercise 5.62, we considered two individuals...Ch. 5.6 - Prob. 85E Ch. 5.6 - Prob. 86E Ch. 5.6 - Prob. 87E Ch. 5.6 - Prob. 88E Ch. 5.7 - In Exercise 5.1, we determined that the joint...Ch. 5.7 - Prob. 90E Ch. 5.7 - In Exercise 5.8, we derived the fact that...Ch. 5.7 - Prob. 92E Ch. 5.7 - Suppose that, as in Exercises 5.11 and 5.79, Y1...Ch. 5.7 - Prob. 94E Ch. 5.7 - Prob. 95E Ch. 5.7 - Prob. 96E Ch. 5.7 - The random variables Y1 and Y2 are such that E(Y1)...Ch. 5.7 - Prob. 98E Ch. 5.7 - Prob. 99E Ch. 5.7 - Let Z be a standard normal random variable and let...Ch. 5.7 - Prob. 101E Ch. 5.8 - A firm purchases two types of industrial...Ch. 5.8 - Prob. 103E Ch. 5.8 - Prob. 104E Ch. 5.8 - Prob. 105E Ch. 5.8 - In Exercise 5.9, we determined that...Ch. 5.8 - In Exercise 5.12, we were given the following...Ch. 5.8 - If Y1 is the total time between a customers...Ch. 5.8 - In Exercise 5.16, Y1 and Y2 denoted the...Ch. 5.8 - Suppose that Y1 and Y2 have correlation...Ch. 5.8 - Prob. 111E Ch. 5.8 - In Exercise 5.18, Y1 and Y2 denoted the lengths of...Ch. 5.8 - A retail grocery merchant figures that her daily...Ch. 5.8 - For the daily output of an industrial operation,...Ch. 5.8 - Prob. 115E Ch. 5.8 - Prob. 116E Ch. 5.8 - A population of N alligators is to be sampled in...Ch. 5.8 - Prob. 118E Ch. 5.9 - A learning experiment requires a rat to run a maze...Ch. 5.9 - Prob. 120E Ch. 5.9 - Refer to Exercise 5.117. Suppose that the number N...Ch. 5.9 - The weights of a population of mice fed on a...Ch. 5.9 - Prob. 123E Ch. 5.9 - The typical cost of damages caused by a fire in a...Ch. 5.9 - When commercial aircraft are inspected, wing...Ch. 5.9 - Prob. 126E Ch. 5.9 - Prob. 127E Ch. 5.10 - Let Y1 and Y2 have a bivariate normal...Ch. 5.10 - Prob. 129E Ch. 5.10 - Prob. 130E Ch. 5.10 - Prob. 131E Ch. 5.10 - Prob. 132E Ch. 5.11 - Prob. 133E Ch. 5.11 - Prob. 134E Ch. 5.11 - In Exercise 5.41, we considered a quality control...Ch. 5.11 - In Exercise 5.42, the number of defects per yard...Ch. 5.11 - In Exercise 5.38, we assumed that Y1, the weight...Ch. 5.11 - Assume that Y denotes the number of bacteria per...Ch. 5.11 - Prob. 139E Ch. 5.11 - Prob. 140E Ch. 5.11 - Let Y1 have an exponential distribution with mean ...Ch. 5.11 - Prob. 142E Ch. 5.11 - Prob. 143E Ch. 5 - Prove Theorem 5.9 when Y1 and Y2 are independent...Ch. 5 - Prob. 145SE Ch. 5 - Prob. 146SE Ch. 5 - Two friends are to meet at the library. Each...Ch. 5 - Prob. 148SE Ch. 5 - Prob. 149SE Ch. 5 - Prob. 150SE Ch. 5 - The lengths of life Y for a type of fuse has an...Ch. 5 - In the production of a certain type of copper, two...Ch. 5 - Suppose that the number of eggs laid by a certain...Ch. 5 - In a clinical study of a new drug formulated to...Ch. 5 - Prob. 155SE Ch. 5 - Refer to Exercise 5.86. Suppose that Z is a...Ch. 5 - Prob. 157SE Ch. 5 - Prob. 158SE Ch. 5 - Prob. 159SE Ch. 5 - Prob. 160SE Ch. 5 - Suppose that we are to observe two independent...Ch. 5 - Prob. 162SE Ch. 5 - Prob. 163SE Ch. 5 - Prob. 164SE Ch. 5 - Prob. 165SE Ch. 5 - Prob. 166SE Ch. 5 - Prob. 167SE

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Solution Summary: The author explains the conditional distribution and density function of Y_12, which are two continuous real valued random variables with joint probability mass function.

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