   Chapter 14.3, Problem 67E

Chapter
Section
Textbook Problem

ProbabilityFind k such that the function f ( x , y ) = { k e − ( x 2 + y 2 ) , x ≥ 0 ,   y ≥ 0 0 , elsewhere is a probability density function.

To determine

To Calculate: The value of k such that f(x,y)={0,elsewhereke(x2+y2),x0,y0 is a probability distribution function

Explanation

Given:

The expression f(x,y)={0,elsewhereke(x2+y2),x0,y0

Formula Used:

For a function f(x,y) to be a probability distribution function

f(x,y)dydx=1

Calculation:

We can write the given problem in double integral as:

00ke(x2+y2)dydx

Now, convert the double integral into polar coordinates by substituting:

x=rcosθy=rsinθdxdy=rdrdθ

The corresponding limits in polar coordinates will be:

0θ

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