4-3 CUMULATIVE DISTRIBUTION FUNCTIONS 103
and
Finally,
Therefore,
The plot of F(x) is shown in Fig. 4-6.
Notice that in the definition of F(x) any can be changed to and vice versa. That is,
F(x) can be defined as either 0.05xor 0 at the end-point and F(x) can be defined as
either 0.05xor 1 at the end-point In other words, F(x) is a continuous function. For a
discrete random variable, F(x) is not a continuous function. Sometimes, a continuous random
variable is defined as one that has a continuous cumulative distribution function.
EXAMPLE 4-4 For the drilling operation in Example 4-2, F(x) consists of two expressions.
for
and for
Therefore,
Figure 4-7 displays a graph of F(x).
F 1 x 2 e
0 x12.5
1 e^201 x12.5^2 12.5x
1 e^201 x12.5^2
F 1 x 2
x
12.5
20 e^201 u12.5^2 du
12.5x
F 1 x 2 0 x12.5
x20.
x0,
F 1 x 2 •
0 x 0
0.05x 0 x 20
120 x
F 1 x 2
x
0
f 1 u 2 du1, for 20 x
F 1 x 2
x
0
f 1 u 2 du0.05x, for 0 x 20
Figure 4-6 Cumulative distribution
function for Example 4-3.
20
1
0 x
F(x)
Figure 4-7 Cumulative distribution
function for Example 4-4.
12.5
1
0 x
F(x)
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