Explanation Let Y be a continuous random variable with density function f ( y ) if y ≥ 0 . If F (y) is the distribution function, show that E ( Y ) = ∫ 0 ∞ y f ( y ) d y = ∫ 0 ∞ [ 1 − F ( y ) ] d y . E ( Y ) = ∫ 0 ∞ y f ( y ) d y = ∫ 0 ∞ ( ∫ 0 y d t ) f ( y ) d y =

Mathematical Statistics with Applications

7th Edition

ISBN: 9780495110811

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

Publisher: Cengage Learning

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Chapter 4.3, Problem 34E

To determine

Prove that E(Y)=∫0∞yf(y)dy=∫0∞[1−F(y)]dy.

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Chapter 4.3, Problem 33E

Chapter 4.3, Problem 35E

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Scenario Sales of products by color follow a peculiar, but predictable, pattern that determines how many units will sell in any given year. This pattern is shown below Product Color 1995 1996 1997 Red 28 42 21 1998 23 1999 29 2000 2001 2002 Unit Sales 2003 2004 15 8 4 2 1 2005 2006 discontinued Green 26 39 20 22 28 14 7 4 2 White 43 65 33 36 45 23 12 Brown 58 87 44 48 60 Yellow 37 56 28 31 Black 28 42 21 Orange 19 29 Purple Total 28 42 21 49 68 78 95 123 176 181 164 127 24 179 Questions A) Which color will sell the most units in 2007? B) Which color will sell the most units combined in the 2007 to 2009 period? Please show all your analysis, leave formulas in cells, and specify any assumptions you make.

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Chapter 4 Solutions

Mathematical Statistics with Applications

Ch. 4.2 - Prob. 1E Ch. 4.2 - A box contains five keys, only one of which will...Ch. 4.2 - A Bernoulli random variable is one that assumes...Ch. 4.2 - Let Y be a binomial random variable with n = 1 and...Ch. 4.2 - Suppose that Y is a random variable that takes on...Ch. 4.2 - Consider a random variable with a geometric...Ch. 4.2 - Let Y be a binomial random variable with n=10 and...Ch. 4.2 - Prob. 8E Ch. 4.2 - A random variable Y has the following distribution...Ch. 4.2 - Refer to the density function given in Exercise...

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Prove that E ( Y ) = ∫ 0 ∞ y f ( y ) d y = ∫ 0 ∞ [ 1 − F ( y ) ] d y .

Solution Summary: The author proves the distribution function E(Y)=displaystyle int

Prove that E(Y)=∫0∞yf(y)dy=∫0∞[1−F(y)]dy.

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Chapter 4 Solutions