a. Explanation Calculation: Let U 1 , U 2 , ... , U n be independent random variables and let V be independent random variable of all U i . The random variables U i and V have the same density function ( f ) and the cdf F . The joint density function of V and U ( n ) is f V , U ( n ) ( v , x ) = f V ( v ) f U ( n ) ( x ) (1) The cumulative distribution function of U ( n ) is F U ( n ) ( x ) = P ( U ( n ) ≤ x ) = P ( U 1 ≤ x , U 2 ≤ x , ... , U n ≤ x ) = P ( U 1 ≤ x ) ︸ F ( x ) P ( U 2 ≤ x ) ︸ F ( x ) ... P ( U n ≤ x ) ︸ F ( x ) = F ( x ) F ( x ) ... F ( x ) = [ F ( x ) ] n Here, the condition is that v ≤ x and v , x ∈ ℝ . This can be written as − ∞ < v ≤ x < ∞ . If v ∈ ℝ is fixed, then x ∈ [ v , ∞ ) b. To determine Find P ( U ( 1 ) ≤ V ≤ U ( n ) ) .

Mathematical Statistics and Data Analysis

3rd Edition

ISBN: 9781111793715

Author: John A. Rice

Publisher: Cengage Learning

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Question

Chapter 3.8, Problem 81P

To determine

Find P(V≤U(n)).

To determine

Find P(U(1)≤V≤U(n)).

Expert Solution & Answer

Chapter 3 Solutions

Mathematical Statistics and Data Analysis

Ch. 3.8 - Prob. 1P Ch. 3.8 - Prob. 2P Ch. 3.8 - Prob. 3P Ch. 3.8 - Prob. 4P Ch. 3.8 - Prob. 5P Ch. 3.8 - Prob. 6P Ch. 3.8 - Prob. 7P Ch. 3.8 - Prob. 8P Ch. 3.8 - Prob. 9P Ch. 3.8 - Prob. 10P

Ch. 3.8 - Prob. 11P Ch. 3.8 - Prob. 12P Ch. 3.8 - Prob. 13P Ch. 3.8 - Prob. 14P Ch. 3.8 - Prob. 15P Ch. 3.8 - Prob. 16P Ch. 3.8 - Prob. 17P Ch. 3.8 - Prob. 18P Ch. 3.8 - Prob. 19P Ch. 3.8 - Prob. 20P Ch. 3.8 - Prob. 22P Ch. 3.8 - Prob. 23P Ch. 3.8 - Prob. 24P Ch. 3.8 - Prob. 25P Ch. 3.8 - Prob. 27P Ch. 3.8 - Prob. 28P Ch. 3.8 - Prob. 29P Ch. 3.8 - Prob. 30P Ch. 3.8 - Prob. 31P Ch. 3.8 - Prob. 32P Ch. 3.8 - Prob. 33P Ch. 3.8 - Prob. 34P Ch. 3.8 - Prob. 35P Ch. 3.8 - Prob. 38P Ch. 3.8 - Prob. 39P Ch. 3.8 - Prob. 44P Ch. 3.8 - Prob. 45P Ch. 3.8 - Prob. 46P Ch. 3.8 - Prob. 47P Ch. 3.8 - Prob. 48P Ch. 3.8 - Prob. 50P Ch. 3.8 - Prob. 51P Ch. 3.8 - Prob. 52P Ch. 3.8 - Prob. 53P Ch. 3.8 - Prob. 54P Ch. 3.8 - Prob. 55P Ch. 3.8 - Prob. 56P Ch. 3.8 - Prob. 57P Ch. 3.8 - Prob. 58P Ch. 3.8 - Prob. 60P Ch. 3.8 - Prob. 61P Ch. 3.8 - Prob. 62P Ch. 3.8 - Prob. 63P Ch. 3.8 - Prob. 64P Ch. 3.8 - Prob. 65P Ch. 3.8 - Prob. 66P Ch. 3.8 - Prob. 67P Ch. 3.8 - Prob. 68P Ch. 3.8 - Prob. 69P Ch. 3.8 - Prob. 70P Ch. 3.8 - Prob. 71P Ch. 3.8 - Prob. 72P Ch. 3.8 - Prob. 73P Ch. 3.8 - Prob. 74P Ch. 3.8 - Prob. 75P Ch. 3.8 - Prob. 76P Ch. 3.8 - Prob. 77P Ch. 3.8 - Prob. 78P Ch. 3.8 - Prob. 79P Ch. 3.8 - Prob. 80P Ch. 3.8 - Prob. 81P

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Find P ( V ≤ U ( n ) ) .

Find P(V≤U(n)).

Find P(U(1)≤V≤U(n)).

Chapter 3 Solutions