On Tue, 5 Nov 2013 07:26:18 -0800 (PST), email@example.com wrote:
>On Monday, November 4, 2013 7:26:07 PM UTC-5, Rich Ulrich wrote: >>On Mon, 4 Nov 2013 10:03:47 -0800 (PST), paul wrote: >> How are curves obtained? Industries probably observed the >> picture before they described it mathematically. Ball-bearings. >> Light-bulbs. Lots of experience. Set in your numbers and >> simulate it. > >I have to cite a reference for completeness because I allude to it in a report.
I was never very good at remember citations for stuff that seems pretty easy and obvious.
Here is a treatment that should be easy enough to understand; and HERE IT IS, posted online. Look at the headers for the message ID. It can be recovered through Google-groups.
Consider a simple system where the time to one failure is
t1= N(10,1); thus for failure 2, 3, ... the SD increases linearly, and the variance as the square of the number t2= N(20,4) t3= N(30,9) .. t25= N(250, 625)
If you plot the first three of them, you will see that there if very, very little overlap between "second failure" for a device and "first failure", and not much for "third failure". But the peak is lower at each successive failure, for that unit of time, and the spread about each peak is increasing in width. For looking at the times of 1, 2, or 3 failures, you can have a pretty exact picture by simply overlaying the three curves.
_________9________5________3___ ... (Number: height of peak)
Depending on how particular you are, you can't just overlay the graphs to show "all failures" after the 3rd or 4th failure; you want to add the densities.
Clearly, by the time you have reached the time of the 25th failure on-the-average, many devices will be on an earlier or later count, because the SD is now 25, and the peak is no longer prominent.