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  • Exponential

    I think it was exponential. Anyway, the problem gave three things. All three did the same thing and were turned on at the same time. The mean was some number, say 5. The idea is that you come back in 3 years and they want the probability that 1 thing is still doing the thing it was doing when it was turned on. It did not say exactly one thing or atleast one thing. It just said 1 thing is still working. So, how do I approach the problem? To me, if 1 thing is still working, then the other 2 are not. Either 1 stopped in year one, 2stopped in year 1, zero stopped in year one, one stopped in year two, or two stopped in year 2. But, this approach did not give me the right answer. I need guidance.

    Thanks.

  • #2
    I think you are asking problems that deal with minimum and maximum of random variable. Usually there are 2 ways that the question can ask:

    1. A machine has 3 identical components. If one of them is spoiled, then it doesn't work anymore.

    Since if one of them expires and the whole machine is gone, we need to find the minimum of 3 random variables for the 3 components. Assuming that they are independent,

    P{min(X1, X2, X3) > 3}
    = exp(-3L) ^3, remember that exp(-3L) is the 1-F(3) of exponential distribution.

    2. A machine has 3 identical components. As long as one of them is working, the machine works.

    We want to find the maximum life span of the component amongst the three. So,

    P{max(X1, X2, X3) < 3}
    = (1- exp(-L)) ^3, remember that 1- exp(-3L) is the F(3) of exponential distribution.
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    • #3
      It was more like you have three things such as clocks. The clocks are all turned on at the same time. You come back in x years, what is the prob that 1 is still ticking?

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      • #4
        I would approach this as 1-P(none are still ticking). So if there are three objects and the mean is 5, after 3 years, this would be 1-(1-e^-.6)^3 = .9082
        Last edited by MKR; June 11 2009, 02:50 PM. Reason: clarification

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        • #5
          I would interpret "1 is still running" as "1 and only 1 is still running".

          MKR, I think that's the probability that the longest running clock runs longer than three years, but doesn't take into account that the other two must fail before 3 years. All three could fail after three years and the max would be > 3.

          I tried this as the probability that a clock fails before 3 years (.451) and the probability that a clock last longer than 3 years 1-.451, and made it binomial:

          (3 choose 1) * (.549)^1 * (.451)^2 = .33

          Maybe I'm all wet, though.

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          • #6
            I would interpret 1 is still running as 1 is running, regardless of the condition of the other two -- unless it said only one is running, or exactly on is running, etc. But yes, it would be helpful to know how exactly the question is worded.

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