Question

A company has figured out how to manufacture a device that runs until one of its three components fail. The lifetimes (in weeks), X1,X2,X3, of these components are independent and identically distributed with an exponential distribution with a mean of eight. Respond to the following questions. (a) What is the probability the device fails within 3 weeks? (b) A third company improves the product by enabling it to function as long as one of the components is still active. Assume that component one is initially on, component two comes on when component one fails, and component three comes on when component two fails. Thus, define the lifetime of the product to be Y = X1 + X2 + X3. What is the MGF of Y ? (State the distribution.) (c) Using your answer from part (b), determine the probability that the device fails within 3 weeks.

Answer 1

a) The life time of the 3 components in this case is .

The CDF of is

The probability that the device fails in 3 weeks is

b) Here . We know the MGF of exponential distribution is

MGF of is

c)

is Gamma distributed with pdf (Here ).

The probability

The R commands for the above integral is

> f <- function (y){y^3*(exp(-y/8))/24576}
> integrate (f,lower=0,upper=3)
0.0006115858 with absolute error < 6.8e-18