Introduction to Probability and Statistics for Engineers and Scientists

4.3Jointly Distributed Random Variables 103

SOLUTION We start by determining the distribution function ofX/Y. Fora> 0

FX/Y(a)=P{X/Y≤a}

=

∫∫

x/y≤a

f(x,y)dx dy

=

∫∫

x/y≤a

e−xe−ydx dy

=

∫∞

0

∫ay

0

e−xe−ydx dy

=

∫∞

0

(1−e−ay)e−ydy

=

[

−e−y+

e−(a+1)y

a+ 1

]∣

∣∣∞

0

= 1 −

1

a+ 1

Differentiation yields that the density function ofX/Yis given by

fX/Y(a)=1/(a+1)^2 ,0<a<∞ ■

We can also define joint probability distributions fornrandom variables in exactly
the same manner as we did forn=2. For instance, the joint cumulative probability
distribution functionF(a 1 ,a 2 ,...,an)ofthenrandom variablesX 1 ,X 2 ,...,Xnis defined
by

F(a 1 ,a 2 ,...,an)=P{X 1 ≤a 1 ,X 2 ≤a 2 ,...,Xn≤an}

If these random variables are discrete, we define their joint probability mass function
p(x 1 ,x 2 ,...,xn)by

p(x 1 ,x 2 ,...,xn)=P{X 1 =x 1 ,X 2 =x 2 ,...,Xn=xn}

Further, thenrandom variables are said to be jointly continuous if there exists a function
f(x 1 ,x 2 ,...,xn), called the joint probability density function, such that for any setCin
n-space

P{(X 1 ,X 2 ,...,Xn)∈C}=

∫∫

(x 1 ,...,xn)∈C

...

∫

f(x 1 ,...,xn)dx 1 dx 2 ···dxn