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Continuous Distribution
Functions
Jake Blanchard
Spring 2010
Uncertainty Analysis for Engineers
1
The Normal Distribution





This is probably the most famous of all
distributions
At one time, many felt that this was the
underlying distribution of nature and that it
would govern all measurements
It is also called the Gaussian distribution
Many random variables are not wellrepresented by this distribution, so its
popularity is not always warranted
Since limits are +/- infinity, this distribution is
problematic in some situations
Uncertainty Analysis for Engineers
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For Example




Suppose we measure the height of many
people and want to represent the data with
a normal distribution
Obviously, the distribution will predict a
finite probability for negative heights, which
makes no sense
On the other hand, a height of 0 will be
several standard deviations from the mean,
so the error will be negligible
In some cases, we can just truncate the
predictions
Uncertainty Analysis for Engineers
3
Normal distribution
mean  
0.45
0.40
0.35
sigma=1,mu=0
std  
3  0
sigma=1, mu=2
sigma=1.2, mu=0
0.30
0.25
f(x) 0.20
4
3
2

0.15
0.10
0.05
0.00
-6.00
-4.00
-2.00
0.00
x
2.00
4.00
6.00
2

1
 x    
f ( x) 
exp 

2
2 
 2

  x  
Uncertainty Analysis for Engineers
4
Central Limit Theorem
One of the key reasons the normal
distribution is the CLT
 It states that the distribution of the mean
of n independent observations from any
distribution with finite mean and variance
will approach a normal distribution for
large n

Uncertainty Analysis for Engineers
5
Examples

Here are some examples of phenomena
that are believed to follow normal
distributions
◦
◦
◦
◦
◦
Particle velocities in a gas
Scores on intelligence tests
Average temperatures in a particular location
Random electrical noise
Instrumentation error
Uncertainty Analysis for Engineers
6
The Half-Normal Distribution

Useful when we are interested in
deviations from the mean
f ( x) 
2
2
  x2 
exp  2 
 2 
0 x
Uncertainty Analysis for Engineers
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Half-Normal Distribution
1.8
1.6
Var(x)=0.5
1.4
Var(x)=1
f(x)
1.2
Var(x)=2
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
x
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When would we use this?
Suppose we build a flywheel from two
parts.
 It is important that they be nearly the
same weight
 We measure only the difference in weight
 This will be positive and is likely to be
normally distributed, with the bulk of the
results near 0
 The half-normal distribution should work

Uncertainty Analysis for Engineers
9
Bivariate Normal Distribution

Joint distribution
1
f ( x, y ) 

4 2 x2 y2 1   2


 1
exp 
2
2
1





 x    2 2  x    y     y    2  
x
y
y
x

  




   
 x y
  x 
y

  

 
E ( x) 
  xf ( x, y)dxdy  
x
  
E ( y)   y


Var ( x)  E x   x    x2
2
Var ( y )   y2

Cov( x, y )
 x y
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What if x and y are not correlated?
The correlation coefficient will be 0
 The joint pdf becomes the product of
two separate normal distributions and x
and y can be considered independent
 Be careful, lack of correlation does not
always imply independence

Uncertainty Analysis for Engineers
11
The Gamma Distribution

For random variables bounded at one end

 x
f ( x) 
 1
x
exp 

  
 
x,  ,   0

    x 1e  x dx
0
Peak is at x=0 for  less than or equal to 1
 CDF is known as incomplete gamma
function

Uncertainty Analysis for Engineers
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Gamma distribution
mean   
Var   
Uncertainty Analysis for Engineers
2
13
Facts on Gamma Distribution
Appropriate for time required for a total
of exactly  independent events to take
place if events occur at a constant rate
1/
 Has been used for storm durations, time
between storms, downtime for offsite
power supplies (nuclear)

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Examples
If a part is ordered in lots of size  and demand
for individual parts is 1/ , then time between
depletions is gamma
 System time to failure is gamma if system failure
occurs as soon as exactly  sub-failures have
taken place and sub-failures occur at the rate 1/ 
 The time between maintenance operations of an
instrument that needs recalibration after  uses is
gamma under appropriate conditions
 Some phenomena, such as capacitor failure and
family income are empirically gamma, though not
theoretically

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Practice

A ferry boat departs for a trip across a
river as soon as exactly 9 cars are loaded.
Cars arrive independently at a rate of 6
per hour. What is probability that the time
between consecutive trips will be less
than one hour? What is the time between
departures that has a 1% probability of
being exceeded?
Uncertainty Analysis for Engineers
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Solution
Time between departures is gamma.
 =9 cars, 1/  =6 per hour
 Evaluate F(1) numerically

1
69
8
F (1) 
t
exp( 6t )dt

(9) 0

Matlab
◦
◦
◦
◦
◦
gamcdf(1,9,1/6)
F=0.153
Or
f=@(x) x.^8.*exp(-6*x)
6^9/gamma(9)*quad(f,0,1)
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Solution continued

Solve this for x
x
69
8 6t
0.99 
t
e dt

(9) 0
gaminv(0.99,9,1/6)
 Solution is x=2.9 hours
 That is, the chances are 1 in 100 that the
time between departures will exceed 2.9
hours

Uncertainty Analysis for Engineers
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Generalized gamma distribution

We can redefine the gamma distribution
to be 0 below some value ()
  
 x 
 1


 

x


exp
x

 

 ( )
f ( x)  

0
elsewhere


Uncertainty Analysis for Engineers
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Exponential Distribution

This is just a gamma distribution with =1
and =1/
Uncertainty Analysis for Engineers
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Exponential distribution
mean 
std 
1

1

f ( x)   exp  x 0  x  
F ( x)  1  exp  x
Uncertainty Analysis for Engineers
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Facts (Exp Distribution)

Useful for time interval between
successive, random, independent events
that occur at constant rates
◦ Time between equipment failures, accidents,
storms, etc.

Given our discussion of the gamma
distribution, this distribution is a good
model for the time for a single outcome
to take places if events occur
independently at a constant rate
Uncertainty Analysis for Engineers
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Example
If particles arrive independently at a
counter at a rate of 2 per second, what is
the probability that a particle will arrive in
1 second?
 =2
 F(1)=1-exp(-2*1)=0.865

Uncertainty Analysis for Engineers
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Beta Distributions

This is useful when x is bounded on both
ends

1  x
f ( x) 
0  x 1
B1 ,  2 
1   2 
B1 ,  2  
1   2 
x
1 1
 2 1
1 ,  2  0
x is bounded between 0 and 1
 f can be u-shaped, single-peaked, J-shaped,
etc.
 The CDF is the incomplete beta function

Uncertainty Analysis for Engineers
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Beta distribution
1
mean 
1   2
1 2
Var 
1   2 2 1   2  1
Uncertainty Analysis for Engineers
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Facts (Beta Distribution)





The many shapes this distribution can take
on make it quite versatile
Often used to represent judgments about
uncertainty
Can be used to represent fraction of time
individuals spend engaging in various
activities
…or fraction of time soil is available for
dermal contact by humans (as opposed to
being covered by soil and ice)
…or fraction of time individual spends
indoors
Uncertainty Analysis for Engineers
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More Examples
A measuring device allows the lengths of
only the shortest and longest units in a
sample to be recorded. 15 units are
selected at random from a large lot. What
is the probability that at least 90% have
lot lengths between the recorded values?
 20 electron tubes are tested until, at time
t, the first one fails. What is the probability
that at least 75% of the tubes will survive
beyond t? Here, 1=1 and 2=0

Uncertainty Analysis for Engineers
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Uniform Distribution

Actually a special case of the beta
distribution (1=1 and 2=1)
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Uniform distribution
ab
mean 
2
2

b  a
Var 
12
1
f ( x) 
ba
a xb
Uncertainty Analysis for Engineers
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Lognormal Distribution

The natural log of the random variable
follows a normal distribution
2

1
 ln( x)    
f ( x) 
exp 

2
2
2  x


0 x

It can be modified to be 0 before some
non-zero value of x
Uncertainty Analysis for Engineers
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Lognormal Distribution
It can be used as a model for a process
whose value results from the
multiplication of many small errors in a
manner similar to the addition of many
instances we discussed with respect to
the normal distribution
 The product of n independent, positive
variates approaches a log-normal
distribution for large n

Uncertainty Analysis for Engineers
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Lognormal distribution
1 

mean   x  exp     2 
2 

Var   x2 exp  2  1
   
Uncertainty Analysis for Engineers
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Lognormal facts

Good for
◦ chemical concentrations in the environment,
deterioration of engineered systems, etc.
◦ asymmetric uncertainties
◦ processes where observed value is random
proportion of previous value

It is “tail-heavy”
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Examples
Distribution of personal incomes
 Distribution of size of organism whose
growth is subject to many small impulses,
the effect of each being proportional to
the instantaneous size
 Distribution of particle sizes from
breakage

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Statistical Models in Life Testing
Time-to-failure models are a common
application of probability distributions
 We can define a hazard function as

f (t )
h(t ) 
1  F (t )
where f and F are the pdf and CDF for
the time to failure, respectively
 h(t)dt represents the proportion of items
surviving at time t that fail at time t+dt

Uncertainty Analysis for Engineers
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Hazard Functions
A typical hazard function is the so-called
bathtub curve, which is high at the
beginning and end of the life cycle
 Uniform distribution – U(0,1)

1
h(t ) 
1 t

Exponential distribution
h(t ) 
e
e
 t
 t

Probability of failure during a
specified interval is constant
Uncertainty Analysis for Engineers
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Weibull Distribution

This is a generalization of the exponential
distribution, but, for time-to-failure
problems, the probability of failure is not
constant
 1
  xL

f ( x)  
  
  x  L  
 
exp  
    
 ,   0; x  L
  x  L  
 
F ( x)  1  exp  
    
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Weibull distribution
 1
mean   1  
 
  2
 1 
Var   2 1     2 1  
  
  
1.4
alpha=1, beta=1, L=0
1.2
3, 1, 0
1.0
f(x)
1, 3, 0
0.8
1, 1, 1
0.6
0.4
0.2
0.0
0.00
0.50
1.00
x
1.50
2.00
2.50
3.00
Uncertainty Analysis for Engineers
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Weibull Facts
Useful for time to completion or time to
failure
 Can skew negative or positive
 Less tail-heavy than lognormal

Uncertainty Analysis for Engineers
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Extreme Value Distributions
Here we are interested in the distribution
of the “largest” or “smallest” element in a
group
 For example,

◦ What is the largest wind gust an airplane can
expect?
Uncertainty Analysis for Engineers
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Types of EV Distributions
Type I (Gumbel) for maximum values
 Type1 (Gumbel) for minimum values
 Type III (Weibull) for minimum values

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The Gumbel Distribution

Limiting model as n approaches infinity
for the distribution of the maximum of n
independent values from an initial
distribution whose right tail is unbounded
and is “exponential”
◦ original distribution could be exponential,
normal, lognormal, gamma, etc. – all have
proper characteristics
Uncertainty Analysis for Engineers
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Type I EV

Can represent
◦ Time to failure of circuit with n elements in
parallel
◦ Yearly maximum of daily water discharges for
a particular river at a particular point
◦ Yearly maximum of the Dow Jones Index
◦ Deepest corrosion pit expected in a metal
exposed to a corrosive liquid for a given time?
Uncertainty Analysis for Engineers
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Type I (Gumbel)

  x  L 
  x  L  
f ( x)  exp 
exp  exp 




 

 


 x  L 
1
h(t )  exp 


  
1
Uncertainty Analysis for Engineers
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Gumbel distribution
mean  L  0.577216
Var 
2
6
2
Uncertainty Analysis for Engineers
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Example
The maximum demand for electric power
at any time during a year in a given
locality is related to extreme weather
conditions
 Assume it follows a Type 1 distribution
with L=2000 kW and =1000 kW.
 A power station needs to know the
probability that demand will exceed 4000
kW at any time in a year and the demand
that has only a 1/20 probability of being
exceeded in a year

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Solution
evcdf(4000,2000,1000)
 =0.9994
 For second part, solve F(y)=0.95 for y
 evinv(0.95,2000,1000)
 Result is 3097 kW

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Type III
This is the Weibull distribution
 It is the limiting model as n approaches
infinity for the distribution of the
minimum of n values from various
distributions bounded at the left
 The gamma distribution is an example
 For example, a circuit with components in
series with individual failure distributions
that are gamma, then the Type III EV
distribution is appropriate

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Other examples
Failure strength of materials
 Drought analyses

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Observations
The log of the weibull distribution is
distributed as a minimum value Type I
 These extreme value distributions are
only valid in the limit of large n –
convergence depends on initial
distributions

◦ 10 samples can be adequate for initial
distributions that are exponential
◦ It may take as many as 100 for normal
distributions
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