Extra life left after the first bulb dies

Question

Two identical bulbs burn independently. Each has an exponential lifetime with a mean of 1000 hours. At some moment the first bulb burns out.

Given that the first bulb has just failed, what is the expected additional time until the second bulb also fails?

Accepted Answer

The exponential distribution is memoryless: however long the surviving bulb has already been lit, its remaining life still has the same exponential distribution it started with. So at the instant the first bulb fails, the second bulb's remaining lifetime is again exponential with a mean of 1000 hours: E[	ext{extra time}] = 1000 	ext{ hours}. The past burn time carries no information; a used exponential bulb is "as good as new." Takeaway. After the first of several exponentials occurs, the wait to the next one is fresh, thanks to memorylessness, and here equals the full single-bulb mean.

Extra life left after the first bulb dies

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