Light bulb question.

**universalsandbox** · January 27th 2009, 06:24 AM

The lifetime of a light bulb is exponentially distributed
mean = 100 hours
Ten bulbs are tested at the same time, and all fail

What is the expected life time of the light bulb that fails first?
What is the probability that the test will last for at least 150 hours?

**mr fantastic** · January 27th 2009, 07:08 PM

Originally Posted by universalsandbox

The lifetime of a light bulb is exponentially distributed
mean = 100 hours
Ten bulbs are tested at the same time, and all fail

What is the expected life time of the light bulb that fails first?
What is the probability that the test will last for at least 150 hours?

Do you mean "Ten bulbs are tested at the same time until all fail"?

What is the expected life time of the light bulb that fails first?

Are you familiar with order statistics? You want the mean value of $\min \{X_1, \, X_2, \, X_3, \, .... \, X_{10}\}$ , that is, $E(X_{(1)})$ .

What is the probability that the test will last for at least 150 hours?

$1 - \Pr(\text{all fail at less than 150 hours}) = 1 - [\Pr(X < 150)]^{10}$ (assuming independence).

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