Cross Product Sample Spaces
497
Table 8.1 Link Status Sample Spaces
Sample Space Link Probability Density Function
021
AB
pl(
0 ) = 0.1, pi(I) = 0.9
022
BC
P2(0) = 0.
2 , P2() = 0.
8
023
AC
p3(0) = 0.5, p3() = 0.
5
the outcome (0, 1, 1) means that AB is down, BC is up, and AC is up. This outcome is
assigned the probability
p(A, 1, 1) = Pl(0) • P2() • p3(1)
= (0.1)(0.8)(0.5)
= 0.04
Figure 8.3 shows an enumeration of the entire sample space QŽ1 x ý22 x Q3 and gives
the values of the cross product probability density function p.
0.5
up (1, 1, 1) (.9)(.8)(.5) = .36
0.80.5
down (1, 1, 0) (.9)(.8)(.5) =.36
0.5
up
(1, 0, 1) (.9)(.2)(.5) =.09
0.5 down (1, 0, 0) (.9)(.2)(.5) =.09
0.5
up (0, 1, 1) (.1)(.8)(.05) = .04
0.5 down (0, 1, 0) (.1)(.8)(.5) =.04
0.5
up (0,0, 1) (.1)(.2)(.5) = .01
0.2
down
0.5 d
(0, 0, 0) (.1)(.2)(.5) = .01
AB
BC
AC
1.00
Figure 8.3 Tree diagram for the communication network.
The event "A is connected to C" occurs when AB and BC are both up; it also occurs
when AC is up. The set E representing this event is
E = {(1, 1, 1), (1, 1, 0), (1, 0, 1), (0, 1, 1), (0,0, 1))