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Non Constant Hazard Rates - Calculating the modal failure rate of hard drive

  1. Dec 21, 2011 #1
    Hi, I'm currently trying to work through a problem about calculating the most likely time for a hard disk to fail:

    Hard disks fail with a probability per unit time: [itex]\alpha (t) = \alpha _0 t [/itex] where [itex]\alpha_0 = 0.5[/itex] years.

    I know that the answer is [itex]t_{modal} = \frac{1}{\sqrt{\alpha_0}}[/itex], but am having problems deriving this. Here is what I've done so far:

    The probability distribution can be calculated as follows:

    [itex]f(x) = \alpha (t) e^{-\int \alpha (t) dt} = \alpha (t) e^{-\frac{1}{2} \alpha_0 t^2} [/itex]

    The most likely time for the disk to fail will be when [itex]\frac{df}{dt} = 0 [/itex]. So when

    [itex] 0 = -{\alpha_0}^2 t^2 e^{-\frac{1}{2} \alpha_0 t^2} [/itex]

    This is where I get stuck. Is this the correct approach? Any ideas about how how I might proceed :)

  2. jcsd
  3. Dec 21, 2011 #2

    Stephen Tashi

    User Avatar
    Science Advisor

    Did you use the product rule when you computed this derivative?
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