Question

Failures of a critical machine part due to cyclical vibration has a gamma distribution with a shape parameter of 2.3 and a scale parameter of 2000 operating hours. Find:

(a) the MTTF

(b) the standard deviation of the time to failure

(c) the probability that a machine part will exceed a lifetime of 1500 hours.

PLEASE SHOW HAND CALCULATIONS/MINITAB WORK!

Answer 1

It is known that if X~Gamma(shape=a,scale=b) then  E(X)=ab and Var(X)=ab²

^{a) Here a=2.3 and b=2000. Then}

^{MTTF=E(X)=2.3*2000=4600}
b) SD=square root of (2.3*2000*2000) =3033.15

c) P(a machine part will exceed a lifetime of 1500 hours)=P(X>1500)=1-P(X<=1500)=1- 0.115651=0.884349