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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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BuyFindarrow_forward

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)={exy,x0,y00, elsewhereP(0x1,xy1)

Formula used:

Double integral formula is:

f(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)0 for all (x,y).

f(x,y)dA=1

P[(x,y)R]=Rf(x,y)dA

Now,

The given function is f(x,y)={exy,x0,y00, elsewhere

Here, x0,y0

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

f(x,y)=0 except x0,y0

f(x,y)dA=00exydxdy=0(e

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