Please help with answering this practice problem 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 \sim \text{Exp}(2) $, that $ B \sim \text{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 $. (b) What is $ E[C] $? Hint: the pdf you obtain in part (a) should be of a familiar type. (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).

Answered: A room is lit by two light bulbs which…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Please help with answering this practice problem

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 \sim \text{Exp}(2) $, that $ B \sim \text{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 $.
(b) What is $ E[C] $?
Hint: the pdf you obtain in part (a) should be of a familiar type.
(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).

Expert Solution

Step 1: we first define the given given information then find the pdf of C and E(C)

Given,

A room is lit by two light bulbs bulbs which are connected in series circuit. if either light bulb fails, the room will go dark

Let $A space a n d space B space$ denote the lifetimes (in years) of these two light bulbs .

Assume that :

$A tilde E x p left parenthesis 2 right parenthesis space a n d space space B tilde E x p left parenthesis 3 right parenthesis$

$A space a n d space B$ are independent of each other

(a)

To find the $p r o b a b i l i t y space d e n s i t y space f u n c t i o n space left parenthesis p d f right parenthesis$ of $C$ , we will consider that $C$ is the minimum of $A space a n d space B$ (since if either bulb fails, the room goes dark)

$f_{C} (c) = P (C = c) = P (m i n (A, B) = c)$

Since $A space a n d space B space$ are independent, we can use the fact that the minimum of independent exponential random variables follows an exponential distribution with rate the parameter equal to the sum of the individual rates. Therefore,

$f_{C} (c) = 5 e^{- 5 c}$ for $c greater or equal than 0 space a n d space 0 space o t h e r w i s e$

(b)

the $space e x p e c t e d space v a l u e s space$ of, $C$ denotes as, $E left parenthesis C right parenthesis$ for an exponential distribution with rate $lambda$ is given by $E left parenthesis C right parenthesis equals 1 over lambda$

In this case the rate parameter of $C$ is $5$ (which is the sum of the individual rate $2 plus 3$ ) so,

$E left parenthesis C right parenthesis equals 1 fifth y e a r s$

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