The life length of a particular component is well modeled by

The life length of a particular component is well modeled by a Weibull distribution having shape parameter beta =2.25 The component design is to be modified in order to increase its characteristic life. For what value of the characteristic life parameter theta will the component population have a 1000 hour reliability of 0.995

Solution

R(t)=e^{-(t/theta)^beta}

given R(t)=0.995

t=1000 hr

beta=2.25

applying logarithm on both sides we get

log_e (R(t)) = -(t/theta)^beta

[- ln R(t)]^1/beta =t/theta

theta=t/[- ln R(t)]^1/beta

putting values we get

theta = 1000/[- ln 0.995]^1/2.25

       =1000/0.09501749128

      =10524.3780543 hrs