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 (xi,yj),i,j 1,2,..., with nonzero
probabilities. The jpmf of X and Y is defined by

for all x and y. It is zero everywhere except at points (xi,yj),i,j 1,2,...,
where it takes values equal to the jo int probability P(X xi Y yj). 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:

where pX (x) and pY (y) are now called marginal probability mass functions. We
also have

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 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 pX Y (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 pX Y (x,y) is zero everywhere except at those points
satisfying x y 5 and x,y 0. Invoking the independence of events of
taking successive steps, it follows from Section 3.3 that pX Y (5, 0), the probabil-
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

52 Fundamentals of Probability and Statistics for Engineers

ˆ

pXY…x;y†ˆP…Xˆx\Yˆy†; … 3 : 20 †

ˆ

ˆ\ˆ

0 <p _

XY…xi;yj†^1 ;

X

i

X

j

pXY…xi;yj†ˆ 1 ;

X

i

pXY…xi;y†ˆpY…y†;

X

j

pXY…x;yj†ˆpX…x†;

9

>>

>>>

>>

>>

>>

=

>>>

>>

>>

>>

>>

;

… 3 : 21 †

FXY…x;y†ˆ

i:Xxix

iˆ 1

jX:yjy

jˆ 1

pXY…xi;yj†: … 3 : 22 †

‡ˆ

‡ ˆ