Answer to Question #234053 in Statistics and Probability for blossomqt

Question

The lifetime of a mechanical assembly in a vibration test is exponentially distributed with a mean of 400 hours. 

a.    What is the probability that an assembly on test fails in less than 100 hours? 

b.    What is the probability that an assembly operates for more than 500 hours before failure? 

c.     If an assembly has been on test for 400 hours without a failure, what is the probability of a failure in the next 100 hours? 

Answer 1

Let $X=$ the lifetime of a mechanical assembly in a vibration test: $X\sim Exp(\lambda).$  

Given $\lambda=1/400=0.0025$

a.

$$P(X<100)=1-e^{-0.0025(100)}=0.2212$$

b.

$$P(X>500)=e^{-0.0025(500)}=0.2865$$

c.

$$P(400<X<400+100)$$

$$=1-e^{-0.0025(500)}-(1-e^{-0.0025(400)})$$

$$=e^{-0.0025(400)}-e^{-0.0025(500)}=0.0814$$