Applied Statistics and Probability for Engineers

152 CHAPTER 5 JOINT PROBABILITY DISTRIBUTIONS

EXAMPLE 5-11 Points that have positive probability in the joint probability distribution of three random variables

X 1 , X 2 , X 3 are shown in Fig. 5-5. The range is the nonnegative integers with x 1 x 2  x 3 3.

The marginal probability distribution of X 2 is found as follows.

Furthermore, E(Xi) and V(Xi) for can be determined from the marginal

probability distribution of Xior from the joint probability distribution of as

follows.

X 1 , X 2 ,p, Xp

i1, 2,p, p

P 1 X 2  32 fX 1 X 2 X 3 1 0, 3, 0 2

P 1 X 2  22 fX 1 X 2 X 3 1 1, 2, 0 2 fX 1 X 2 X 3 1 0, 2, 1 2

P 1 X 2  12 fX 1 X 2 X 3 1 2, 1, 0 2 fX 1 X 2 X 3 1 0, 1, 2 2 fX 1 X 2 X 3 1 1, 1, 1 2

P 1 X 2  02 fX 1 X 2 X 3 1 3, 0, 0 2 fX 1 X 2 X 3 1 0, 0, 3 2 fX 1 X 2 X 3 1 1, 0, 2 2 fX 1 X 2 X 3 1 2, 0, 1 2

If are discrete random variables with joint probability mass func-

tion the marginal probability mass function of any Xiis

(5-9)

where Rxidenotes the set of points in the range of (X 1 , X 2 ,, p Xp) for whichXixi.

fXi 1 xi 2 P 1 Xixi 2  a

Rxi

fX 1 X 2 p (^) Xp 1 x 1 , x 2 ,p, xp 2
fX 1 X 2 p (^) Xp 1 x 1 , x 2 ,p, xp 2 ,
X 1 , X 2 , X 3 ,p, Xp
Definition
1
0
023 x 1
1
2
3
x 3
x 2
2
3
Figure 5-5 Joint 1
probability distribution
ofX 1 , X 2 , andX 3.
and
(5-10)
where Ris the set of all points in the range of X 1 , X 2 ,p, Xp.
V 1 Xi 2  a
R
1 xiXi 22 fX 1 X 2 p (^) Xp 1 x 1 , x 2 ,p, xp 2
E 1 Xi 2  a
R
xi fX 1 X 2 p (^) Xp 1 x 1 , x 2 ,p, xp 2
Mean and
Variance from
Joint
Distribution
With several random variables, we might be interested in the probability distribution of some
subset of the collection of variables. The probability distribution of kpcan
be obtained from the joint probability distribution of X 1 , X 2 ,p, Xpas follows.
X 1 , X 2 ,p, Xk,
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