A Room Lit by Two Light Bulbs in a Series Circuit

Probability Distribution and Expected Lifetime of Light Bulbs

A room is lit by two light bulbs which are connected in a series circuit. If either light bulb fails, the room will go dark. Let A and B denote the lifetimes (in years) of these two light bulbs. Assume that A ~ Exp(2), that B ~ Exp(3), and that A and B are independent of each other.

(a) Let C denote the time when the room goes dark. Find the pdf of C.

The pdf of C, the time when the room goes dark, is given by f(c) = 5e^(-5c).

(b) What is E[C]? Hint: the pdf you obtain in part (a) should be of a familiar type.

The expected value of C, E[C], is 0.2 years.

(c) Suppose light bulb B burns out first. You then move light bulb A to another room where it is the only bulb. Starting at the time it is turned on in the new room, what is the expected lifetime of bulb A in the new room? Hint: this question can be answered independently of parts (a) and (b).

The expected lifetime of bulb A in the new room is 0.5 years.

Explanation: The expected lifetime of bulb A in the new room can be calculated independently of parts (a) and (b) since we are only considering the lifetime of bulb A in isolation. Therefore, the expected lifetime of bulb A in the new room remains the same as before, which is 0.5 years.