Continuous Distributions

Normal Distribution

40%

30%

20%

10%

0%

+ + 2 + 3 – – 2 – 3

68%

95%

99.73%

50%

f xx

( ) 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

zx

=−

Normal Parameter Estimation

= = =

x

x

n

i

i

n

1

( )

( ) = =

−

−

=

s

x x

n

i

i

n2

1

1

( )

= =

−

−

= =

s

n x x

n n

i

i

n

i

i

n2

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

−−=

2ln

2

1e xp

2

1)(

x

xxf

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

PVT =

V

tf

dVeQ-

= Vtf

dVeQ-

=

r

AaE

2=

Lognormal Probability Density Function

0 2 4 6 80

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 xxxf 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 20 250

0.02

0.04

0.06

0.08

0 50 100 150 200 2500

0.002

0.004

0.006

0.008

q = 10

q = 100

0 50 100 150 200 250 300 3500

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 slopeShape parameter

• β < 1Failure rate decreases with operating timeProducts having defects emanating from manufacturing, storing or mounting are screened out at an early stage.

• β = 1Failure rate is constantNo 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 timeProcess 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

Decreasingfailure rate

Constantfailure rate

Increasingfailure rate

Fail

ure

ra

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 Fail48

Time to Fail15

Time to Fail97

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(t2t1)=R(t2)/R(t1)

• Example

• If x is exponential then 1/x is Poisson

Estimating Lognormal Parameters

Estimating Weibull Parameters