STAT 528
HOMEWORK #7
Due: Monday, November 16, 2015.
1. (1pt) Show the pth percentile for the two-parameter Weibull distribution is yp = α(−ln(1 − P ))1/β .
2. (1pt) Show the hazard function for the two-parameter Weibull distribution is h(y) = (β/α)(y/α)β−1 .
3. (2pt) Show the hazard function for the two-parameter Weibull distribution increases for β > 1 and
decreases for β < 1.
4. A study of the lifetime for a mechanical switch was performed. 40 switches were put into constant
operation until the time of failure Y (measured in millions of operations). The following table contains
the switch failure results with actual failure times denoted by 1 and censored failure times denoted by 0.
• Try to find a good fitting distribution to Y .
• Try to find a good fitting distribution to ln(Y ).
(a) (5pt) For the two best fitting distributions (based on either Y or ln(Y )) provide the pdfs that
incorporate the associated parameter estimates.
(b) (5pt) Briefly justify why you selected these two distributions. If this involves plots and/or statistics,
then include the plots and/or statistics for these two distributions.
Y
1.151
1.171
1.248
1.331
1.381
1.499
1.508
1.534
1.577
1.584
1.667
1.695
1.710
1.955
1.965
2.012
2.051
2.076
2.109
2.116
ln(Y )
0.14063
0.15786
0.22154
0.28593
0.32281
0.40480
0.41078
0.42788
0.45552
0.45995
0.51103
0.52768
0.53649
0.67039
0.67549
0.69913
0.71833
0.73044
0.74621
0.74953
censor
0
0
0
0
0
1
0
0
0
0
1
1
1
0
1
0
0
0
1
0
Y
2.119
2.135
2.197
2.199
2.227
2.250
2.254
2.261
2.349
2.369
2.547
2.548
2.738
2.794
2.883
2.883
2.910
3.015
3.017
3.793
ln(Y )
0.75094
0.75847
0.78709
0.78800
0.80066
0.81093
0.81271
0.81581
0.85399
0.86247
0.93492
0.93531
1.00723
1.02747
1.05883
1.05883
1.06815
1.10360
1.10426
1.33316
censor
0
1
1
0
1
0
1
0
0
1
1
1
0
1
0
0
1
1
1
0