11.2 Basic Concepts of Probability Theory 639
11.2.5 Jointly Distributed Random Variables
When two or more random variables are being considered simultaneously, their joint
behavior is determined by a joint probability distribution function. The probability
distributions of single random variables are calledunivariate distributions and the
distributions that involve two random variables are calledbivariate distributions. In
general, if a distribution involves more than one random variable, it is called amulti-
variate distribution.
Joint Density and Distribution Functions. We can define thejoint density function
ofncontinuous random variablesX 1 , X 2 ,... , Xnas
fX 1 ,...,Xn(x 1 ,... , xn) dx 1 · · · dxn= P(x 1 ≤X 1 ≤x 1 + dx 1 ,
x 2 ≤X 2 ≤x 2 + dx 2 ,... , xn≤Xn≤xn+ dxn) (11.20)
If the random variables are independent, the joint density function is given by the
product of individual or marginal density functions as
fX 1 ,...,Xn(x 1 ,... , xn)=fX 1 (x 1 ) · ·· fXn(xn) (11.21)
The joint distribution function
FX 1 ,X 2 ,...,Xn(x 1 , x 2 ,... , xn)
associated with the density function of Eq. (11.20) is given by
FX 1 ,...,Xn(x 1 ,... , xn)
=P[X 1 ≤x 1 ,... , Xn≤xn]
=
∫x 1
−∞
· · ·
∫xn
−∞
fX 1 ,...,Xn(x 1 ′, x′ 2 ,... , x′n) dx′ 1 dx′ 2 · · · dxn′ (11.22)
IfX 1 , X 2 ,... , Xnare independent random variables, we have
FX 1 ,...,Xn(x 1 ,... , xn)=FX 1 (x 1 ) FX 2 (x 2 ) · ·· FXn(xn) (11.23)
It can be seen that the joint density function can be obtained by differentiating the joint
distribution function as
fX 1 ,...,Xn(x 1 ,... , xn)=
∂n
∂x 1 ∂x 2 · · · ∂xn
FX 1 ,...,Xn(x 1 ,... , xn) (11.24)
Obtaining the Marginal or Individual Density Function from the Joint Density
Function. Let the joint density function of two random variablesXandYbe denoted
byf(x, y)and the marginal density functions ofXandYbyfX(x) andfY(y) respec-,
tively. Take the infinitesimal rectangle with corners located at the points (x, y), (x+dx,
y), (x, y+dy), and (x+dx,y+dy). The probability of a random point (x′, y′) allingf
in this rectangle isfX,Y ( , y)x dx dy. The integral of such probability elements with
respect toy(for a fixed value ofx)is the sum of the probabilities of all the mutually
