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?
There are a 100 lightbulbs and the first one that will burn out is the minimum of the exponential random variables. Sum the lambdas (rate parameters) to get this minimum’s lambda. All these lambdas have the same value so you can do 100 times lambda, which will give you the lambda for this minimum random variable.
\[ \lambda = 1000 \] hours \[ 100 * \lambda \]
\[ 100 * 1/1000 = 1/10 \]
But this quantity is not in hours, it is the reciprocal. So invert it to get the expected time for the first bulb to burn out, 10 hours.