Life Distribution Mathematical Functions

An important aspect of Reliability Engineering is the analysis of time-to-failure data (also known as life data) using statistical methods to assess the reliability of a population of devices and predict potential failures rates.  Life data are not analyzed as individual numbers, but collectively as a life distribution.  As its name implies, a life distribution shows how a population of devices fail in time, or how the failures are distributed in time.

There are four (4) life distributions generally used in semiconductor reliability analyses today, namely, the normal distribution, the lognormal distribution, the Weibull distribution, and the exponential distribution.  Among these, the lognormal distribution most closely represents most of the failure mechanisms in the semiconductor industry today.

Each of the four life distributions may be described by three important mathematical functions:

1)  the probability density function, f(t), which indicates the relative frequency of failures at any time, t;

2)  the cumulative density function, F(t), which gives the probability that a device will fail at or before time t; and

3) the curve of failure rate,  l(t), which indicates the instantaneous failure rate at any time, t.

Tables 1 to 4 show the important characteristics of the 4 life distributions, particularly their respective probability density functions, cumulative density functions, and instantaneous failure rates.

Table 1. Normal Distribution

 Probability Density Function f(t) = (e^{-0.5[(t-µ)/s]2}) / (sÖ2p) Cumulative Density Function F(t) = (1/[sÖ2p]) ò0t e^{-0.5[(x-µ)/s]2}dx Instantaneous Failure Rate l(t) = e^{-0.5[(t-µ)/s]2}/òt¥ e^{-0.5[(x-µ)/s]2}dx Median t = t50% = µ Mean t = µ Mode t = µ Location Parameter µ Shape Parameter s s - s,estimate of s, may be calculated as t50%-t16%

Table 2. Exponential Distribution

 Probability Density Function f(t) = le-lt Cumulative Density Function F(t) = 1 - e-lt Instantaneous Failure Rate l(t) = f(t)/(1-F(t)) Mean or MTBF t = 1/l

Table 3. Lognormal Distribution

 Probability Density Function f(t) = e^{-0.5[(ln(t)-µ)/s]2} / (stÖ2p) Cumulative Density Function F(t) = (1/[sÖ(2p)]) ò0t (1/x) e^{-0.5[(ln(x)-µ)/s]2}dx Instantaneous Failure Rate l(t) = f(t)/(1-F(t)) Median t = t50% = eµ Mean t = e^(µ+s2/2) Mode t = e^(µ-s2) Location Parameter eµ Shape Parameter s s - s,estimate of s, may be calculated as ln(t50%/t16%)

Table 4. Weibull Distribution

 Probability Density Function f(t) = ([b(t-g)b-1]/[ab]) (e^{-[(t-g)/a]b}) Cumulative Density Function F(t) = 1 - e^{-[(t-g)/a]b} Instantaneous Failure Rate l(t) = [b(t-g)b-1]/[ab] Location Parameter a = t at 63.2% failure Shape Parameter b Time Delay Parameter g, not used unless data do not fit the distribution without time delay