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

1. Dec 21, 2011

### 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

2. Dec 21, 2011

### Stephen Tashi

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