Explanation Calculation: It is given that the random variable P is a uniform distribution on [ 0 , 1 ] and the probability density function is f P ( p ) = 1 , p ∈ [ 0 , 1 ] . The random variable X | P is Bernoulli distribution with parameter p and the probability density function is, f X | P ( x | p ) = P ( X = x | P = p ) = p x ( 1 − p ) 1 − x , x ∈ { 0 , 1 } and p ∈ [ 0 , 1 ] The conditional distribution of P given X is, f P | X ( p | x ) = f X , P ( x , p ) f X ( x ) . Similarly, the conditional distribution of X given P is f X | P ( x | p ) = f X , P ( x , p ) f P ( p ) . By using the conditional distribution of X given P , the joint density function is f X | P ( x | p ) = f X , P ( x , p ) f P ( p ) f X , P ( x , p ) = f X | P ( x | p ) f P ( p ) ︸ = 1 = p x ( 1 − p ) 1 − x ( 1 ) = p x ( 1 − p ) 1 − x ; x ∈ { 0 , 1 } and p ∈ [ 0 , 1 ] The marginal density of X is, f X ( x ) = ∫ 0 1 f X , P ( x , p ) d p = ∫ 0 1 p x ( 1 − p ) 1 − x d p Here, the simplification of marginal density is not required because the random variable X falls in the interval [0, 1]

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 24P

To determine

Find the conditional distribution of P given X.

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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Similar questions

The conditional probability of E given that F occurs is P(EF)=___________. So in rolling a die the conditional probability of the event E, getting a six, given that the event F, getting an even number, has occurred is P(EF)=___________.
The conditional probability of E given that F occur is P(EF)= _____________. So in rolling a die the conditional probability of the event E. “getting a six,” given that the event F, “getting an even number.” has occurred is P(EF)= ____________.
Assume that the probability that an airplane engine will fail during a torture test is 12and that the aircraft in question has 4 engines. Construct a sample space for the torture test. Use S for survive and F for fail.

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