A room has two lamps that use bulbs of type A and B, respectively. The lifetime,

Question

A room has two lamps that use bulbs of type A and B, respectively. The lifetime, X, of any particular bulb of a particular type is a random variable, independent of everything else, with the following PDF: for type-A Bulbs: f_X(x) = { e^-x, if x greaterthanorequalto 0, 0, otherwise: for type-B Bulbs: f_X(x) = {3e^-3x, if x greaterthanorequalto 0, 0, otherwise. Both lamps are lit at time zero. Whenever a bulb is burned out it is immediately replaced by a new bulb. (a) What is the expected value of the number of type-B bulb failures until time t? (b) What is the PDF of the time until the first failure of either bulb type? (c) Find the expected value and variance of the time until the third failure of a type-B bulb. (d) Suppose that a type-A bulb has just failed. How long do we expect to wait until a subsequent type-B bulb failure?

Explanation / Answer

A) Type b follows exponential distribution with parameter 3

Expected value of exponential = 1/parameter. => 1/3

B) PDF(A B) = PDF(A) x PDF(B)

=> 3e-4x  for x>=0

C) Expected value = 1/3

Variance = 1/32  => 1/9

D) A and B are independent

So expectation of B after A = expectation of B

=> 1/3

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