Page 69 - Fundamentals of Probability and Statistics for Engineers

52 Fundamentals of Probability and Statistics for Engineers

discrete. Let X and Y be two discrete random variables that assume at most
a countably infinite number of value pairs (x i , y j ), i, j ˆ 1, 2, .. ., with nonzero
probabilities. The jpmf of X and Y is defined by

p x; y†ˆ P X ˆ x \ Y ˆ y†; 3:20†
XY

for all x and y. It is zero everywhere except at points (x i , y j ), i, j ˆ 1, 2,.. .,
ˆ
where it takes values equal to the joint probability P(X ˆ x i \ Y y j ). We
observe the following properties, which are direct extensions of those noted in
Equations (3.4), (3.6), and (3.7) for the single-random-variable case:

_ 9
0 < p x i ; y j † 1;
XY >
>
>
>
X X >
p x i ; y j †ˆ 1; >
>
XY >
>
i j >
>
=
3:21†
X
p
XY x i ; y†ˆ p y†; >
Y
>
i >
>
>
>
X
>
p XY x; y j †ˆ p x†; >
>
X
>
>
j ;
where p (x) and p (y) are now called marginal probability mass functions. We
X
Y
also have
i:x i x j:y j y
X X
F XY x; y†ˆ p x i ; y j †: 3:22†
XY
iˆ1 jˆ1
Example 3.5. Problem: consider a simplified version of a two-dimensional
‘random walk’ problem. We imagine a particle that moves in a plane in unit
steps starting from the origin. Each step is one unit in the positive direction, with
probability p along the x axis and probability q ( p ‡ q ˆ 1) along the y axis. We
further assume that each step is taken independently of the others. What is the
probability distribution of the position of this particle after five steps?
Answer: since the position is conveniently represented by two coordinates,
we wish to establish p XY (x, y) where random variable X represents the x
coordinate of the position after five steps and where Y represents the y coord-
inate. It is clear that jpmf p XY (x, y) is zero everywhere except at those points
satisfying x ‡ y ˆ 5 and x, y 0. Invoking the independence of events of
taking successive steps, it follows from Section 3.3 that p (5, 0), the probabil-
XY
ity of the particle being at (5, 0) after five steps, is the product of probabilities of
taking five successive steps in the positive x direction. Hence
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