2.13. GOODNESS-OF-FIT TEST:
2
TEST 87
Example 2.45
e failure time X in the hour of a device is an exponential distribution with a failure rate D
0:01
1
hour
. (1) Calculate the mean of the failure time. (2) Calculate probability P .X 25/.
Solution:
1. Calculate the mean of the failure time.
Per Equation (2.123), the mean of the failure time is:
x
D
1
D
1
0:01
D 100 .hours/:
2. P .X 25/.
Using the Microsoft Excel per Equation (2.126), the P .X 25/ is:
P
.
X 25
/
D F
.
25
/
D EXPON:DIST
.
25; 0:01; true
/
D 0:2212:
Using the MATLAB per Equation (2.128), the P .X 25/ is:
P
.
X 25
/
D F
.
x
/
D expcdf
.
25; 1=0:01
/
D 0:2212:
2.13 GOODNESS-OF-FIT TEST:
2
TEST
2.13.1 INTRODUCTION
When enough sample data (at least more than 30) is obtained, the histogram of the collected
data can be drawn. Per the shape of the histogram, we can assume the type of distribution.
For example, several histograms are displayed in Figure 2.27. From the histograms, the random
variable with the histogram of Figure 2.27a might follow a normal distribution. e random
variable with the histogram of Figure 2.27b might follow a lognormal distribution or a Weibull
distribution. e random variable with the histogram of Figure 2.27c might follow an expo-
nential distribution. e random variable with the histogram of Figure 2.27d might follow a
uniform distribution.
e goodness-of-fit test is to determine whether a sample belongs to a hypothesized the-
oretical distribution. It estimates the adequacy of a fit by determining the difference between
the frequency of occurrence of a random variable characterized by an observed sample and the
expected frequency obtained from the hypothesized distribution.
ere are different methods for conducting a goodness-of-fit test. One method is to use
the
2
test with the Excel functions. Another method is to use the MATLAB command for the
goodness-of-fit test, which is still based on the
2
test.
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