Introduction to Probability and Statistics for Engineers and Scientists

5.4The Uniform Random Variable 167

Because

1 =

∫

R

f(x,y)dx dy

=

∫

R

cdxdy

=c×Area ofR

it follows that

c=

1

Area ofR

For any regionA⊂R,

P{(X,Y)∈A}=

∫∫

(x,y)∈A

f(x,y)dx dy

=

∫∫

(x,y)∈A

cdxdy

=

Area ofA

Area ofR

Suppose now thatX,Yis uniformly distributed over the following rectangular regionR:

0, b a, b

0, 0 a, 0

R

Its joint density function is

f(x,y)=

{

c if 0≤x≤a,0≤y≤b

0 otherwise

wherec=Area of rectangle^1 =ab^1. In this case,XandYare independent uniform random

variables. To show this, note that for 0≤x≤a,0≤y≤b

P{X≤x,Y≤y}=c

∫x

0

∫y

0

dy dx=

xy

ab

(5.4.5)