Chapter 14.2, Problem 64E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# In Exercises 61-64, show that the function is a joint density function and find the required probability. f ( x , y ) = { e − x − y , x ≥ 0 ,     y ≥ 0 0 , elsewhere P ( 0 ≤ x ≤ 1 ,   x ≤ y ≤ 1 )

To determine

To calculate: The probability for the function

f(x,y)={exy,x0,y00, elsewhereP(0x1,xy1),

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

Explanation

Given:

The function is:

f(x,y)={eâˆ’xâˆ’y,xâ‰¥0,yâ‰¥00,Â elsewhereP(0â‰¤xâ‰¤1,xâ‰¤yâ‰¤1)

Formula used:

Double integral formula is:

âˆ«âˆ’âˆžâˆžâˆ«âˆ’âˆžâˆžf(x,y)dA=1

Calculation:

A function f(x,y) is said to be a joint density function if the following properties are satisfied:

f(x,y)â‰¥0 for all (x,y).

âˆ«âˆ’âˆžâˆžâˆ«âˆ’âˆžâˆžf(x,y)dA=1

P[(x,y)âˆˆR]=âˆ¬Rf(x,y)dA

Now,

The given function is f(x,y)={eâˆ’xâˆ’y,xâ‰¥0,yâ‰¥00,Â elsewhere

Here, xâ‰¥0,yâ‰¥0

Since exponential function can never be negative and zero, f(x,y)â‰¥0 for all (x,y).

f(x,y)=0 except xâ‰¥0,yâ‰¥0

âˆ«âˆ’âˆžâˆžâˆ«âˆ’âˆžâˆžf(x,y)dA=âˆ«0âˆžâˆ«0âˆžeâˆ’xâˆ’ydxdy=âˆ’âˆ«0âˆž(eâˆ’

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