Explanation Given: The function is: f ( x , y ) = { e − x − y , x ≥ 0 , y ≥ 0 0 , elsewhere P ( 0 ≤ x ≤ 1 , x ≤ y ≤ 1 ) Formula used: Double integral formula is: ∫ − ∞ ∞ ∫ − ∞ ∞ f ( x , y ) d A = 1 Calculation: A function f ( x , y ) is said to be a joint density function if the following properties are satisfied: f ( x , y ) ≥ 0 for all ( x , y ) . ∫ − ∞ ∞ ∫ − ∞ ∞ f ( x , y ) d A = 1 P [ ( x , y ) ∈ R ] = ∬ R f ( x , y ) d A Now, The given function is f ( x , y ) = { e − x − y , x ≥ 0 , y ≥ 0 0 , elsewhere Here, x ≥ 0 , y ≥ 0 Since exponential function can never be negative and zero, f ( x , y ) ≥ 0 for all ( x , y ) . f ( x , y ) = 0 except x ≥ 0 , y ≥ 0 ∫ − ∞ ∞ ∫ − ∞ ∞ f ( x , y ) d A = ∫ 0 ∞ ∫ 0 ∞ e − x − y d x d y = − ∫ 0 ∞ ( e −

11th Edition

ISBN: 9781337275347

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Question

Chapter 14.2, Problem 64E

To determine

To calculate: The probability for the function

f(x,y)={e−x−y,x≥0,y≥00, elsewhereP(0≤x≤1,x≤y≤1),

and to prove that the function is a joint density function.

Expert Solution & Answer

Chapter 14 Solutions

Calculus (MindTap Course List)

Ch. 14.1 - CONCEPT CHECK Iterated Integral Explain what is...Ch. 14.1 - Prob. 2E Ch. 14.1 - Question: Evaluate the integral: 0x(2xy)dy Ch. 14.1 - Evaluating an IntegralIn Exercises 310, evaluate...Ch. 14.1 - Evaluating an IntegralIn Exercises 310, evaluate...Ch. 14.1 - Evaluating an IntegralIn Exercises 310, evaluate...Ch. 14.1 - Evaluating an Integral In Exercises 3-10, evaluate...Ch. 14.1 - Evaluating an IntegralIn Exercises 310, evaluate...Ch. 14.1 - Evaluating an Integral In Exercises 3-10, evaluate...Ch. 14.1 - Evaluating an Integral In Exercises 3-10, evaluate...

Knowledge Booster

The probability for the function f ( x , y ) = { e − x − y , x ≥ 0 , y ≥ 0 0 , elsewhere P ( 0 ≤ x ≤ 1 , x ≤ y ≤ 1 ) , and to prove that the function is a joint density function.

Solution Summary: The author explains that the given function is a joint density function and the required probability is 0.1997.

To calculate: The probability for the function

f(x,y)={e−x−y,x≥0,y≥00, elsewhereP(0≤x≤1,x≤y≤1),

and to prove that the function is a joint density function.

Chapter 14 Solutions

The probability for the function f ( x , y ) = { e − x − y , x ≥ 0 , y ≥ 0 0 , elsewhere P ( 0 ≤ x ≤ 1 , x ≤ y ≤ 1 ) , and to prove that the function is a joint density function.

Solution Summary: The author explains that the given function is a joint density function and the required probability is 0.1997.

To calculate: The probability for the function f(x,y)={e−x−y,x≥0,y≥00, elsewhereP(0≤x≤1,x≤y≤1), and to prove that the function is a joint density function.

Chapter 14 Solutions

To calculate: The probability for the function

f(x,y)={e−x−y,x≥0,y≥00, elsewhereP(0≤x≤1,x≤y≤1),

and to prove that the function is a joint density function.