Solution
The Used Lightbulb
Show the problem again
The lifetime of a certain lightbulb is exponentially distributed with mean 1000 hours. You find one in a hallway that has been burning continuously for exactly 1000 hours and is still working. What is the expected total lifetime of this bulb, measured in hours from when it was first switched on?
Worked solution
The answer is 2000 hours. The exponential distribution is memoryless: P(T > s + t | T > s) = e^(−(s+t)/1000) / e^(−s/1000) = P(T > t). A bulb that has survived 1000 hours is, probabilistically, as good as new, so its expected remaining life is still 1000 hours. Adding the 1000 hours already elapsed gives E[T | T > 1000] = 2000. The tempting answers, "1000, because that is the average" or "nearly zero, because it is due to fail", both treat the bulb as if it wore out. Exponential lifetimes model components that fail by sudden accident rather than by ageing.
Source: Classic exponential-distribution exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.