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

[orangeIV]

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: \alpha (t) = \alpha _0 t where \alpha_0 = 0.5 years.

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

The probability distribution can be calculated as follows:

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

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

0 = -{\alpha_0}^2 t^2 e^{-\frac{1}{2} \alpha_0 t^2}

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

Thanks

 

[Stephen Tashi]

when \frac{df}{dt} = 0. So when

0 = -{\alpha_0}^2 t^2 e^{-\frac{1}{2} \alpha_0 t^2}

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