Introduction to Probability and Statistics for Engineers and Scientists

4.3Jointly Distributed Random Variables 101

SOLUTION

(a) P{X>1,Y< 1 }=

∫ 1

0

∫∞

1

2 e−xe−^2 ydx dy

=

∫ 1

0

2 e−^2 y(−e−x|∞ 1 )dy

=e−^1

∫ 1

0

2 e−^2 ydy

=e−^1 (1−e−^2 )

(b) P{X<Y}=

∫∫

(x,y):x<y

2 e−xe−^2 ydx dy

=

∫∞

0

∫y

0

2 e−xe−^2 ydx dy

=

∫∞

0

2 e−^2 y(1−e−y)dy

=

∫∞

0

2 e−^2 ydy−

∫∞

0

2 e−^3 ydy

= 1 −^23

=^13

(c) P{X<a}=

∫a

0

∫∞

0

2 e−^2 ye−xdy dx

=

∫a

0

e−xdx

= 1 −e−a ■

4.3.1 Independent Random Variables

The random variablesXandY are said to be independent if for any two sets of real
numbersAandB

P{X∈A,Y ∈B}=P{X∈A}P{Y∈B} (4.3.7)

In other words,XandYare independent if, for allAandB, the eventsEA={X ∈A}
andFB={Y∈B}are independent.
It can be shown by using the three axioms of probability that Equation 4.3.7 will follow
if and only if for alla,b

P{X≤a,Y≤b}=P{X≤a}P{Y≤b}