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: fX(x) =

e^−x, if x ≥ 0,

0, otherwise;

for type-B Bulbs: fX(x) =

3e^−3x, if x ≥ 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?

The time is infinity? how is this possible?

(b) What is the PDF of the time until the first failure of either bulb type?

lamdatotal = lamda 1 + lamda2

4te^(-4t) ?

(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?