Explanation Calculation: The Farlie-Morgenstern family is H ( x , y ) = F ( x ) G ( y ) ( 1 + a [ 1 − F ( x ) ] ) [ 1 − G ( y ) ] (1) where, H represents the joint cumulative distribution function, and F and G represents cumulative distribution functions which has marginal cumulative distribution functions of H . Let F ( x ) = 1 − e − λ x and G ( y ) = 1 − e − μ y . Consider F ( x ) is an exponential random variable with parameter λ and G ( y ) is an exponential random variable with parameter μ . Substitute F ( x ) and G ( y ) in Equation (1). H ( x , y ) = ( 1 − e − λ x ) ( 1 − e − μ y ) ( 1 + α [ 1 − ( 1 − e − λ x ) ] ) [ 1 − ( 1 − e − μ y ) ] = ( 1 − e − λ x ) ( 1 − e − μ y ) ( 1 + α [ 1 − 1 + e − λ x ] ) [ 1 − 1 + e − μ ] = ( 1 − e − λ x ) ( 1 − e − μ y ) ( 1 + α e − λ x ) ( e − μ y ) = ( 1 − e − λ x ) ( 1 − e − μ y ) ( 1 + α ( e − λ x − μ y ) ) The joint probability density function is, h ( x , y ) = ∂ 2 ∂ y ∂ x H ( x , y ) = ∂ 2 ∂ y ∂ x [ ( 1 − e − λ x ) ( 1 − e − μ y ) ( 1 + α e − λ x − μ y ) ] = ∂ ∂ x [ ( 1 − e − λ x ) ∂ ∂ y ( ( 1 − e − μ y ) ( 1 + α e − λ x − μ y ) ) ] = ∂ ∂ x [ ( 1 − e − λ x ) ( μ e − μ y ( 1 + α e − λ x − μ y ) − ( 1 − e − μ y ) α μ e − λ x − μ y ) ] Further simplification, h ( x , y ) = ∂ ∂ x

Mathematical Statistics and Data Analysis

3rd Edition

ISBN: 9781111793715

Author: John A. Rice

Publisher: Cengage Learning

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Chapter 3.8, Problem 29P

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Find an expression for the joint density.

Solution Summary: The author calculates the Farlie-Morgenstern family in Equation (1).

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