A company buys 100 lightbulbs, each of which has an exponential lifetime of 1000 hours. What is the expected time for the first of these bulbs to burn out? (See Exercise 10.)
We have 100 Lightbulbs with an expoential lifetime of 1000 hours.
This means that the Expected lifetime of the lightbulbs is 1000 hours: \(E[X]=1000\) hours.
Then, since \(E[X]=1000\) this means that \(\lambda= \frac{1}{1000}\)
From problem 10 we know that the mean of the minimum of the independent random variables of an exponential is \(\frac{\mu}{n}\)
We know that \(n = 100\) and \(\mu=1000\)
So, by solving for this with the mean we will figure out the expected lifetime for the first bulb to run out.
\(E[firstLightBuldBurnOut]=\frac{1000}{10}=10\) hours