I have come across this problem:
A room is lit by $10$ light bulbs. The lifetime, X, of the light bulbs follows an exponential distribution with mean $ \mu$ = $1000$ hours. In a time window of $2000$ hours:
- what is the expected number of failures?
I did this as: For time window of 2000 hours, and 10 bulbs, expected value= $\frac{1}{1000}$ * $2000$ * $10$ = $20$
- what is the probability that there will be more than $5$ failures?
The light bulbs are replaced immediately upon failure.
I calculated the probability that a light bulb will not fail (X> $2000$) is:
$ \mu$ =$1000$ , $\lambda$ = $\frac{1}{1000}$
P(X>2000) = $e^{-\lambda x}$ = $e^{-2000/1000}$ = 0.13 ( 13% )
and tried to incorporate that in to solving the questions but I think I'm heading the wrong way.
Edit: I also used Cumulative Poisson Probability to find upper/lower limits but I don't know how to apply it to multiple objects. Also I am curious as to how the problem would change if the light bulbs were not being replaced upon failure.