Question

Thirty items are placed on a replacement test that is to be operated until 15 failures occur. The fifteenth failure occurs at 100 hours. Estimate the following:

i) The mean time to failure;

ii) 95% two-sided limits on R(120);

iii) The time for which the reliability will be 0.85 with point and two-sided confidence interval estimates.

Answer 1

Let the random variable

X: the time between one failure and the next

then  

Also the random variable
Y: the time until k = 15 failures occurs is then Gamma distributed



Now    = Expected time until 15 failures occur,
Now as the 15th failure occurred at 100 hours, using the method of moments

Estimate the following:
i) the mean time to failure

the mean time to failure = E[X]
an estimate of the mean time of failure =  Ans (i)

ii) 95% two-sided limits on R(120);

An exact 100(1 – α) % confidence interval for the R(t) is given by

where,

is the value of the chi-square random variate with degrees of freedom

Using

,