Applied Statistics and Probability for Engineers

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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