Fundamentals of Probability and Statistics for Engineers

For pX Y (4,1), there are five distinct ways of reaching that position (4 steps in
the x direction and 1 in y; 3 in the x direction, 1 in y, and 1 in the x direction;
and so on), each with a probability of p^4 q. We thus have

Similarly, other nonvanishing values of pX Y (x, y) are easily calculated to be

The jpmf pX Y (x,y) is graphically presented in Figure 3.11 for p 0 4 and
q 0 6. It is easy to check that the sum of pX Y (x, y) over all x and y is 1, as
required by the second of Equations (3.21).
Let us note that the marginal probability mass functions of X and Y are,
following the last two expressions in Equations (3.21),

pXY(x,y)

0.4

0.3

0.2

0.1

0

012345

1

2

3

4

5

x

y

Figure 3. 11 The joint probability mass function, pX Y (x,y), for Example 3.5, with

Random Variables and Probability D istributions 53

pXY… 5 ; 0 †ˆp^5 :

pXY… 4 ; 1 †ˆ 5 p^4 q:

pXY…x;y†ˆ

10 p^3 q^2 ; for…x;y†ˆ… 3 ; 2 †;

10 p^2 q^3 ; for…x;y†ˆ… 2 ; 3 †;

5 pq^4 ; for…x;y†ˆ… 1 ; 4 †;

q^5 ; for…x;y†ˆ… 0 ; 5 †:

8

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ˆ :

ˆ :

pX…x†ˆ

X

j

pXY…x;yj†ˆ

q^5 ; forxˆ 0 ;

5 pq^4 ; forxˆ 1 ;

10 p^2 q^3 ; forxˆ 2 ;

10 p^3 q^2 ; forxˆ 3 ;

5 p^4 q; forxˆ 4 ;

p^5 ; forxˆ 5 ;

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pˆ 0 :4 andqˆ 0 : 6