General Examples Using the Exponential Distribution

Exponential Distribution Example 8

Twenty units were reliability tested with the following results:

Number of Units in Group	Time-to-Failure
7	100
5	200
3	300
2	400
1	500
2	600

8-1. Assuming a two-parameter exponential distribution, estimate the parameters analytically using the MLE method.

8-2. Repeat part 8-1 using Weibull++ (enter the data as grouped data to duplicate the results of 8-1).

8-3. Plot the exponential probability vs. time-to-failure using Weibull++. (Note: Note that you will be plotting MLE results on points placed on the plot using a rank methodology. This rank methodology assumes 7 failures occurred by 100 hours, 12 by 200 and so forth.)

8-4. Plot R(t) vs. time using Weibull++.

8-5. Plot the pdf using Weibull++.

8-6. Plot the failure rate vs. time using Weibull++.

8-7. Estimate the parameters analytically using the RRY method (using grouped ranks).

Solution To Exponential Distribution Example 8

8-1. For the two-parameter exponential distribution and for = 100 hours (first failure), the partial of the log-likelihood function, Λ, becomes:

8-2. The data as entered in Weibull++ along with results are shown next.

8-3. The exponential probability plot is shown next.

8-4. The Special Plot drop-down box allows you to select different plot types, shown next.

Select Reliability vs. Time.

8-5. The exponential pdf plot is shown next.

8-6. The exponential failure rate plot is shown next.

Note that, as described at the beginning of the Exponential Distribution chapter, the failure rate for the exponential distribution is constant. Also note that the failure rate plot does not exist for times before the location parameter, γ, at 100 hours.

8-7. In the case of grouped data, one must be cautious when estimating the parameters using a rank regression method. That is because the median rank values are determined from the total number of failures observed by time Ti where i indicates the group number. In this example the total number of groups is N = 6 and the total number of units is NT = 20. Thus, the median rank values will be estimated for twenty units and for the total failed units (NFi) up to the ith group, for the ith rank value. The median ranks values can be found from rank tables or they can be estimated using ReliaSoft's Quick Statistical Reference . For example, the median rank value of the fourth group will be the 17th rank out of a sample size of twenty units (or 81.945% ). (Note: Note that all data points can be treated as individual values in Weibull ++ by selecting the Use all data if grouped option on the Set Analysis page of the Folio Control Panel.)

The following table is then constructed (as in Example 2).

Given the values in the table above, calculate and using Eqns. (7) and (8):

or:

and:

or:

Therefore, from Eqn. (6):

and from Eqn. (5):

or:

Then:

Using Weibull++ , the estimated parameters are:

The small difference in the values from Weibull++ is due to rounding. In Weibull++ the calculations and the rank values are carried out up to the 15th decimal point.

Exponential Distribution Example 9

A number of leukemia patients were treated with either drug 6 MP or a placebo, and the times in weeks until cancer symptoms returned were recorded. Analyze each treatment separately. [21, p.175]

Table 7.3 - Leukemia Treatment Results

Time (weeks)	Number of Patients	Treatment	Comments
1	2	placebo
2	2	placebo
3	1	placebo
4	2	placebo
5	2	placebo
6	4	6MP	3 patients completed
7	1	6MP
8	4	placebo
9	1	6MP	Not completed
10	2	6MP	1 patient completed
11	2	placebo
11	1	6MP	Not completed
12	2	placebo
13	1	6MP
15	1	placebo
16	1	6MP
17	1	placebo
17	1	6MP	Not completed
19	1	6MP	Not completed
20	1	6MP	Not completed
22	1	placebo
22	1	6MP
23	1	placebo
23	1	6MP
25	1	6MP	Not completed
32	2	6MP	Not completed
34	1	6MP	Not completed
35	1	6MP	Not completed

Solution to Exponential Distribution Example 9

Enter the data into Weibull++, by selecting Times to Failure, with Right Censored Data (Suspensions) and with Grouped Observations. In the first column enter the number of patients. Whenever there are uncompleted tests, enter the number of patients who completed the test separately from the number of patients who did not. In the second column enter F for completed tests and S for uncompleted. In the third column enter the time. In the fourth column (Subset ID) enter the name of the treatment. The title of each column can be changed by double-clicking it and typing the desired name.

Now click the Batch Auto Run icon,

and click Select All Available> > to separate the 6 MP drug from the placebo as shown next.

Click OK and you will get a new Data Sheet for each treatment with the corresponding results, as shown next.

From the Project menu, click Add Additional Plot, then choose Add Multiplot. Click the Select Data Sheets button

and select the two data sheets under the Folio1 project.

The plot is shown next.