Fundamentals of Probability and Statistics for Engineers

[see Equation (2.26)], for three events A, B, and C, we have, in the case of three
random variables X, Y, and Z,

Hence, for the general case of n random variables, X 1 ,X 2 ,...,Xn,or X,wecan
write

In the event that these random variables are mutually independent, Equations
(3.49) become

Example 3.9.To show that joint mass functions are sometimes more easily
found by finding first the conditional mass functions, let us consider a traffic
problem as described below.
Problem: a group of n cars enters an intersection from the south. Through
prior observations, it is estimated that each car has the probability p of turning
east, probability q of turning west, and probability r of going straight on
(p q r 1). Assume that drivers behave independently and let X be the
number of cars turning east and Y the number turning west. Determine the
jpmf pX Y (x, y).
Answer: since

we proceed by de te rmining pX Y (xy) and pY (y). The marginal mass function
pY (y) is found in a way very similar to that in the random walk situation
described in Example 3.5. Each car has two alternatives: turning west, and
not turning west. By enumeration, we can show that it has a binomial distribu-
tion (to be more fully justified in Chapter 6)

64 Fundamentals of Probability and Statistics for Engineers

pXYZ…x;y;z†ˆpXYZ…xjy;z†pYZ…yjz†pZ…z†

fXYZ…x;y;z†ˆfXYZ…xjy;z†fYZ…yjz†fZ…z†

9

=

;

… 3 : 48 †

pX…x†ˆpX 1 X 2 ...Xn…x 1 jx 2 ;...;xn†pX 2 ...Xn…x 2 jx 3 ;...;xn†...pXn 1 Xn…xn 1 jxn†pXn…xn†;

fX…x†ˆfX 1 X 2 ...Xn…x 1 jx 2 ;...;xn†fX 2 ...Xn…x 2 jx 3 ;...;xn†...fXn 1 Xn…xn 1 jxn†fXn…xn†:

9

=

;

… 3 : 49 †

pX…x†ˆpX 1 …x 1 †pX 2 …x 2 †...pXn…xn†;

fX…x†ˆfX 1 …x 1 †fX 2 …x 2 †...fXn…xn†:

)

… 3 : 50 †

‡‡ˆ

pXY…x;y†ˆpXY…xjy†pY…y†;

j

pY…y†ˆ

n

y



qy… 1 q†ny; yˆ 1 ; 2 ;...: … 3 : 51 †