# 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**.