(a) A lamp has two bulbs of a type with an average lifetime of 1600 hours. Assum

Question

(a) A lamp has two bulbs of a type with an average lifetime of 1600 hours. Assuming that we can model the probability of failure of these bulbs by an exponential density function with mean %u03BC = 1600, find the probability that both of the lamp's bulbs fail within 1700 hours. (Round your answer to four decimal places.)

(b) Another lamp has just one bulb of the same type as in part (a). If one bulb burns out and is replaced by a bulb of the same type, find the probability that the two bulbs fail within a total of 1700 hours. (Round your answer to four decimal places.)

Explanation / Answer

It's an exponential density function, so it is of the form p = Ae^(-kt), where t is the time in hours and k and A are positive constants.

The total must be 1, so (integral from 0 to infinity) Ae^(-kt) = 1
=> [from 0 to infinity] -A/k e^(-kt) = 1
=> A/k e^0 = 1
=> A/k=1 or A = k
So we rewrite it p = ke^(-kt).

Now the mean is 1600,
=> average of f(t) times t is 1600,
=> integral of f(t) times t is 1600 (since total area under f(t) is 1)
=> (integrate t from 0 to infinity) kt e^(-kt) = 1600

You should be able to integrate kt e^(-kt), and so find k, after which it's all pretty straightforward.

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