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686
Probability and Random Variables
Appendix B
P (x)
(a)
A
D
B
C
F
G
x ( )
Random variable
(maps events into values
on the real line)
0.3
0.2
0.1
E
H
Venn diagram
3 2 1 0
1 2 3
x
Event
[·]
Value of
Random Variable
x[·]
Probability of Event
P(x)
A 0.0 0.10
B -3.0 0.05
C -1.5 0.20
D -2.0 0.15
E +0.5 0.10
F +1.0 0.10
G +2.0 0.00
H +3.0 0.30
Total = 1.00
Figure B–2
Random variable and probability functions for Example B–2.
DEFINITION. The probability density function (PDF) of the random variable x is given
by f(x), where
f(x) = dF(a)
da
2 =
a = x
where f(x) has units of 1x.
dP(x … a)
2
da a = x
= lim
n: q
¢x:0
l
c
¢x an ¢x
n bd
(B–15)
Example B–2 (Continued)
The CDF for this example that was illustrated in Fig. B–2 is easily obtained using Eq. (B–14).
The resulting CDF is shown in Fig. B–3. Note that the CDF starts at a zero value on the
left (a =-q), and that the probability is accumulated until the CDF is unity on the right
(a =+q).