a. Explanation It is given that a random variable Y has the distribution function . F ( y ) = { 0 if y < 0 y 2 + 0.1 if 0 ≤ y < 0.5 y if 0 .5 ≤ y < 1 1 if y ≥ 1 Denote F 1 ( y ) and F 2 ( y ) the discrete and continuous components of F ( y ) . Now, define the distribution function F ( y ) is as follows: F ( y ) = c 1 F 1 ( y ) + c 2 F 2 ( y ) ∀ y Taking y → ∞ , then c 1 + c 2 = 1 . Since F 2 ( y ) is continuous, then F ( 0 ) − lim y → 0 − F ( y ) = c 1 ( F 1 ( 0 ) − lim y → 0 − F 1 ( y ) ) 0.1 − 0 = c 1 ( F 1 ( 0 ) − 0 ) { ∵ F 1 ( y ) = 0 if y < 0 } c 1 F 1 ( 0 ) = 0.1 F ( 0.5 ) − lim y → 0.5 − F ( y ) = c 1 ( F 1 ( 0.5 ) − lim y → 0.5 − F 1 ( y ) ) 0.5 − 0.35 = c 1 ( F 1 ( 0.5 ) − F 1 ( 0 ) ) c 1 F 1 ( 0.5 ) = 0.15 + 0.1 Since F 1 ( y ) = F 1 ( 0 ) ∀ y ∈ [ 0 , 0 b. To determine Express F ( y ) as c 1 F 1 ( y ) + c 2 F 2 ( y ) . c. To determine Find the expected value and variance of 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 4.11, Problem 159E

To determine

Find F1(y) and F2(y), the discrete and continuous components of F(y).

To determine

Express F(y) as c1F1(y)+c2F2(y).

To determine

Find the expected value and variance of Y.

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Chapter 4.11, Problem 158E

Chapter 4, Problem 160SE

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Find F 1 ( y ) and F 2 ( y ) , the discrete and continuous components of F ( y ) .

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Find F1(y) and F2(y), the discrete and continuous components of F(y).

Express F(y) as c1F1(y)+c2F2(y).

Find the expected value and variance of Y.

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Chapter 4 Solutions