Life
Distribution Mathematical Functions
An important
aspect of Reliability Engineering is the analysis
of timetofailure 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[(tµ)/s]^{2}})
/ (sÖ2p) 
Cumulative Density Function

F(t) =
(1/[sÖ2p])
ò_{0}^{t}
e^{0.5[(xµ)/s]^{2}}dx 
Instantaneous Failure Rate

l(t)
= e^{0.5[(tµ)/s]^{2}}/ò_{t}^{¥}
e^{0.5[(xµ)/s]^{2}}dx 
Median

t = t_{50%}
= µ 
Mean

t = µ 
Mode

t = µ 
Location Parameter

µ 
Shape
Parameter
s

s

s,estimate of
s, may be
calculated as t_{50%}t_{16%} 
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)/(1F(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)])
ò_{0}^{t}
(1/x)
e^{0.5[(ln(x)µ)/s]^{2}}dx 
Instantaneous Failure Rate

l(t)
= f(t)/(1F(t)) 
Median

t = t_{50%}
= e^{µ} 
Mean

t = e^(µ+s^{2}/2) 
Mode

t = e^(µs^{2}) 
Location Parameter

e^{µ} 
Shape
Parameter
s

s 
s,estimate of
s,
may be calculated as ln(t_{50%}/t_{16%}) 
Table 4. Weibull Distribution
Probability Density Function

f(t) = ([b(tg)^{b1}]/[a^{b}])
(e^{[(tg)/a]^{b}}) 
Cumulative Density Function

F(t) =
1  e^{[(tg)/a]^{b}} 
Instantaneous Failure Rate

l(t)
= [b(tg)^{b1}]/[a^{b}]

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 
See
also:
separate article on
Life Distributions
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