A light fixture contains five lightbulbs. The lifetime of each bulb is exponentially

Chapter 4, Problem 11

(choose chapter or problem)

A light fixture contains five light bulbs. The lifetime of each bulb is exponentially distributed with mean 200 hours. Whenever a bulb burns out, it is replaced. Let  be the time of the first bulb replacement. Let \(X_{i}, i=1, \ldots, 5\), be the lifetimes of the five bulbs. Assume the lifetimes of the bulbs are independent.

a. Find \(P\left(X_{1}>100\right)\)

b. Find \(P\left(X_{1}>100 \text { and } X_{2}>100 \text { and } \cdots \text { and } X_{5}>100\right)\)

c. Explain why the event \(T>100\) is the same as \(\left\{X_{1}>100 \text { and } X_{2}>100 \text { and } \cdots \text { and } X_{5}>100\right\}\)

d. Find \(P(T \leq 100)\).

e. Let  be any positive number. Find \(P(T \leq t)\), which is the cumulative distribution function of .

f. Does  have an exponential distribution?

g. Find the mean of .

h. If there were  lightbulbs, and the lifetime of each was exponentially distributed with parameter , what would be the distribution of  ?

Equation Transcription:

   

   

   

   

   

Text Transcription:

Xi, i=1,...,5  

P(X1>100)    

P(X1>100 and X2>100 and \cdots and X5>100)

T>100    

{X1>100 and X2>100 and \cdots and X5>100}    

P(T \leq 100)    

P(T \leq t)