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Continuous_Distributions1.pdf
Continuous Distributions
Normal Distribution
40%
30%
20%
10%
0%
+ + 2 + 3 - - 2 - 3
68%
95%
99.73%
50%
f x x
( ) exp= − −
1
2
1
2
2
There is no closed solution for the integral of the normal probability density function.
The standard normal random deviate (z) was introduced to allow integration via tables
• Mean of z =0 • STS of z = 1
This is no longer relevant, but the term z is still used
z x
= −
Normal Parameter Estimation
= = =
x
x
n
i
i
n
1
( )
( ) = =
−
−
=
s
x x
n
i
i
n 2
1
1
( )
= =
−
−
= =
s
n x x
n n
i
i
n
i
i
n 2
1 1
2
1
Requires only a single pass for computer code
Normal Distribution
Properties:
1.Symmetrical
2.Measures of central tendency are all identical
(mean, median, mode, and midrange)
Central Limit Theorem • Sums & averages become
Normal
• Y = x1 + x2 + … + xn
• Y = (x1 + x2 + … + xn)/n
• Regardless of the distribution of the individuals
• Excel Example
Lognormal Distribution
• If a data set is known to follow a lognormal distribution, transforming the data by taking a logarithm yields a data set that is normally distributed.
• Limited to right skewed data
− −=
2 ln
2
1 e xp
2
1 )(
x
x xf
Lognormal Data Normal Data
12 ln(12)
16 ln(16)
28 ln(28)
48 ln(48)
87 ln(87)
143 ln(143)
Lognormal Distribution
• Y = (x1)(x2) … (xn)
• ln(Y) = ln(x1) + ln(x2) + … + ln(xn)
• When a system is the result of multiplication or division the result tends to be lognormal
• Lognormal distribution in engineering
• Ideal gas law
• Salt in a tank with flow & mixing
• Electrical field (coaxial capacitor)
Lognormal Distribution
nR
PV T =
V
tf
dVeQ -
= V tf
dVeQ -
=
r
Aa E
2 =
Lognormal Probability Density Function
0 2 4 6 8 0
0.1
0.2
0.3
0.4
0.5
0.6
x
f(x)
=
= =
=
=
=
Weibull Probability Density Function
• b = shape parameter
• q = scale parameter
• d = location parameter
− −
− =
− b
b
b
dqdq
b xx xf e xp
)( )(
1
0
0
X
f(x) b=
b=
b=
b=
b=
q
Effects of the Shape Parameter
Effects of the Scale Parameter
0 5 10 15 20 25 0
0.02
0.04
0.06
0.08
0 50 100 150 200 250 0
0.002
0.004
0.006
0.008
q = 10
q = 100
0 50 100 150 200 250 300 350 0
0.002
0.004
0.006
0.008
Effects of the Location Parameter
d = 100
Weibull F(x) & R(x)
b
q
d
− −
−=
x
exF 1)( b
q
d
− −
=
x
exR )(
Bathtub curve
• The bathtub curve is widely used in reliability engineering. It describes a particular form of the hazard function which comprises three parts:
• "infant mortality"
• "useful life"
• "wear out"
• The bathtub curve is generated by mapping the rate of early "infant mortality" failures when first introduced, the rate of random failures with constant failure rate during its "useful life", and finally the rate of "wear out" failures as the product exceeds its design lifetime.
• h(t) - Hazard function
Weibull slope Shape parameter
• β < 1Failure rate decreases with operating time Products having defects emanating from manufacturing, storing or mounting are screened out at an early stage.
• β = 1Failure rate is constant No memory of previous stress history, i.e. an old, still running product may be as good as a fresh one.
• β > 1Failure rate increases with operating time Process indicates deterioration of material properties (structure).
Bathtub curve
h(L)
Life (time)
β < 1 β = 1 β > 1
Early "infant mortality" failures Wear out
failures
Decreasing failure rate
Constant failure rate
Increasing failure rate
Fa il
u re
r a
te
Normal life
Weibull Hazard function
X
h(x)
b = 0.5
b = 1
b = 2
b = 3.6 b = 9
1/q
h(t) - Hazard function • Instantaneous failure rate
• A measure of proneness to failure as a function of the age of units
Extreme values
Time to Fail 48
Time to Fail 15
Time to Fail 97
The system fails when one of the 3 components fail. What is the system time to fail?
The system fails when all of the 3 components fail. What is the system time to fail?
Exponential distribution
• Constant failure rate
• Lack of memory • R(t2\t1)=R(t2)/R(t1)
• Example
• If x is exponential then 1/x is Poisson
Estimating Lognormal Parameters
Estimating Weibull Parameters
