Introduction to Probability and Statistics for Engineers and Scientists

100 Chapter 4:Random Variables and Expectation

it follows, upon differentiation, that

f(a,b)=

∂^2

∂a∂b

F(a,b)

wherever the partial derivatives are defined. Another interpretation of the joint density
function is obtained from Equation 4.3.4 as follows:

P{a<X<a+da,b<Y<b+db}=

∫d+db

b

∫a+da

a

f(x,y)dx dy

≈f(a,b)da db

whendaanddbare small andf(x,y) is continuous ata,b. Hencef(a,b) is a measure of
how likely it is that the random vector (X,Y) will be near (a,b).
IfX andY are jointly continuous, they are individually continuous, and their
probability density functions can be obtained as follows:

P{X∈A}=P{X∈A,Y∈(−∞,∞)} (4.3.5)

=

∫

A

∫∞

−∞

f(x,y)dy dx

=

∫

A

fX(x)dx

where

fX(x)=

∫∞

−∞

f(x,y)dy

is thus the probability density function ofX. Similarly, the probability density function
ofYis given by

fY(y)=

∫∞

−∞

f(x,y)dx (4.3.6)

EXAMPLE 4.3c The joint density function ofXandYis given by

f(x,y)=

{

2 e−xe−^2 y 0 <x<∞,0<y<∞

0 otherwise

Compute(a)P{X>1,Y< 1 };(b)P{X<Y}; and(c)P{X<a}.