Page 81 - Fundamentals of Probability and Statistics for Engineers

64 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 ,

9
p x; y; z†ˆ p xjy; z†p
Z
XYZ XYZ YZ yjz†p z† =
3:48†
f x; y; z†ˆ f xjy; z†f yjz†f z† ;
XYZ XYZ YZ Z
Hence, for the general case of n random variables, X 1 , X 2 ,..., X n , or X, we can
write

9
p x†ˆ p X 1 X 2 ...X n x 1 jx 2 ;...;x n †p X 2 ...X n x 2 jx 3 ;...;x n †...p X n 1 X n x n 1 jx n †p x n †; =
X
X n
f x†ˆ f X 1 X 2 ...X n x 1 jx 2 ;...;x n †f X 2 ...X n x 2 jx 3 ;...;x n †...f X n 1 X n x n 1 jx n †f X n x n †: ;
X
3:49†
In the event that these random variables are mutually independent, Equations
(3.49) become

p x†ˆ p x 1 †p x 2 † ... p x n †; )
X
X 1
X 2
X n
3:50†
f x†ˆ f X 1 x 1 †f X 2 x 2 † ... f X n x n †:
X
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
r
q
( p ‡ ‡ ˆ 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 p XY (x, y).
Answer: since

p x; y†ˆ p xjy†p y†;
XY XY Y
we proceed by determining p XY (x y) and p (y). The marginal mass function
j
Y
p (y) is found in a way very similar to that in the random walk situation
Y
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)

n y n y
p y†ˆ q 1 q† ; y ˆ 1; 2; ... : 3:51†
Y
y

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