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        riddles >> medium >> Lightbulb Race
        
(Message started by: william wu on Feb 9^th, 2006, 2:01am)

Title: Lightbulb Race
Post by william wu on Feb 9^th, 2006, 2:01am

The lifetime of a Brand A lightbulb is distributed as an exponential random variable with parameter 2. A Brand B lightbulb's lifetime is distributed as an exponential random variable with parameter 1.

You have two options: to buy both brand A bulbs and chain their lifetimes together (meaning you replace the first bulb with the second as soon as it goes out), or just buy the brand B bulb.

What is the probability that the brand B bulb outlives two brand A bulbs?

What happens in the limit, if Brand A's parameter is n times the parameter of Brand B's?

Note: This problem can be solved with rote dirty integration of conditional densities, but there exists a very simple and clean solution.

Source: John Gill

Title: Re: Lightbulb Race
Post by desi on Feb 9^th, 2006, 2:11pm

In the limit
[hideb]1/e[/hideb]

Title: Re: Lightbulb Race
Post by desi on Feb 9^th, 2006, 2:16pm

Explanation
[hideb]
If X(1), X(2) ... X(n) are i.i.d exp r.v's with mean 1.
A B bulb has life X(i)/n
n B bulbs have life Z(n)=(X(1) + ... X(n))/n ==> a.s. 1
Hence, A - Z(n) ==> a.s A - 1
P(A > Z(n) ) ==> P(A > 1) = 1/e

[/hideb]