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MEGN 537 – Probabilistic Biomechanics
Ch.4 – Common Probability Distributions
Anthony J Petrella, PhD
Common Terms
• Random Variable: A numerical description of an
experimental outcome. The domain (sometimes
called the “range”) is the set of all possible values
for the random variable
• Probability Distribution: A representation of all the
possible values of a random variable and the
corresponding probabilities.
Continuous and Discrete Probability
Distributions
• Probability Distributions can be continuous or discrete based on the
type of values contained within the domain of the random variable.
Normal or Gaussian Distribution
• Frequently, a stable, controlled process will produce a
histogram that resembles the bell shaped curve also
known as the Normal or Gaussian Distribution
• The properties of the normal distribution make it a highly utilized
distribution in understanding, improving, and controlling processes
Common applications:
Astronomical data
Exam scores
Human body temperature
Human birth weight
Dimensional tolerances
Financial portfolio management
Employee performance
Normal Distribution
• Continuous Data
• Typically 2 parameters
• Scale parameter = mean (mx)
• Shape parameter = standard deviation (sx)
• PDF
 1  x  m 2 
x 
fx ( x) 
exp  
s x 2
 2  s x  
• CDF
 1  x  m 2 
x   dx
Fx ( x) 
exp  
 2  s x  
 s x 2
1

x
1
Normal Distribution
Distributions and Probability
• Distributions can be linked to probability – making possible
predictions and evaluations of the likelihood of a particular
occurrence
• In a normal distribution, the number of standard deviations from
the mean tells us the percent distribution of the data and thus
the probability of occurrence
Standard Normal Distribution
CDF
PDF
0.5
m=0
s=1
0.4
0.3
0.8
F(x)
f(x)
1
0.6
0.2
0.4
0.1
0.2
0
-6.0
-4.0
-2.0
0.0
x
2.0
4.0
6.0
0
-6.0
-4.0
-2.0
0.0
x
2.0
4.0
6.0
Standard Normal Distribution
• Normal (m=0, s=1)
• Standard normal variate
• (Note: Halder uses S)
x  mx
z
sx
• All normal distributions can be simply transformed
to the standard normal distribution
• Probability
probabilit y  ( z )  F ( z )

z(b)
 1 2
P(a  x  b) 
exp  s ds  (z(b))  (z(a))
 2 
z(a)
Probability for other Sigma
Values?
• Suppose we want to calculate the amount of data included at
X < 2.65s (Probability at 2.65s from the mean)
• How will we figure out the area for such a particular standard
deviation measurement?
The probability density function is:
 1  x  m 2 
x 
fx ( x) 
exp  
s x 2
 2  s x  
1
For given values of X, m and s we
could calculate the area under the
curve, however, it would be unwise
to go through this process every
time we need to make a calculation
The Standard Normal Distribution
Negative z Values
( z )  1  ( z )
Solving for (z)
• There is no closed form solution for the CDF
of a normal distribution
• Common solution methods
• Use a look-up table
• Use a software package (Excel, SAS, etc.)
• Perform numerical integration (e.g. apply
trapezoidal or Simpson’s 1/3 rule)
Experimental Data
• Fitting a distribution to the experimental data
• Determine m and s
• Use these as the distribution parameters
• Plot the raw data together with the normal
curve representation and evaluate whether
the distribution is normally distributed
Normal Distributions in Excel
General distributions
• norm.dist(x,mean,stdev,cumulative) – returns
a probability at the specified value of the variable
• cumulative = true (1) for CDF, cumulative = false (0) for PDF
• norm.inv(p,mean,stdev) – returns the value of the
variable at the specified probability level
Standard normal distributions
• norm.s.dist(z,cumulative) – returns probability
• norm.s.inv(p) – returns the value of the std normal
variate, z
Means and Tails
• What aspects of data are most interesting from
an engineering standpoint? Extreme conditions
• Highest temperature or stress
• Shortest life to failure
• Understanding the tails of a distribution can be
critical to understanding performance
• It is difficult to collect data in the tails
 distribution allows you to maximize data
Remember this is an assumption!
Lognormal Distribution
• Natural log (ln) of the random variable has a
normal distribution
2

1
1  ln(x)   x 
 
fx ( x) 
exp  
 x 2x
x
 2 
 
• Determination of lognormal parameters from
mean and standard deviation
1 2
 x  E(ln(x ))  ln m x -  x
2
  s 2 
 x2  Var(ln(x))  ln 1   x  
  m x  
Lognormal Distribution
• Common applications:
• Fatigue life to failure
• Material Strength
• Loading spectra
0.5
0.5
m=3
s=1
0.3
0.4
f(x)
f(x)
0.4
0.3
0.2
0.2
0.1
0.1
0
0
50
100
150
x
200
250
0
-2.0
0.0
2.0
ln(x)
4.0
6.0
Lognormal Distribution
2

1
1  ln(x)   x 
 
fx ( x) 
exp  
 x 2x
x
 2 
 
where =scale and  = shape
Lognormal Distribution
ln x   x
z
x
• Standard Normal Variate, z:
• Probability:

z(b)
 1 2
P(a  x  b) 
exp  s ds
 2 
z(a)
 (z(b))  (z(a))
Important Features
• From Haldar, p.71
• If X is a lognormal variable with parameters
x and x, then ln(X) is normal with a mean
of x and a standard deviation of x
• When COV, dx ≤ 0.3 x ≈ dx,
Lognormal Distributions in Excel
General distributions
• lognorm.dist(x,mean,stdev,cumulative) –
returns the probability
• cumulative = true for CDF, cumulative = false for PDF
• lognorm.inv(p,mean,stdev) – returns the value of
the variable
Transform with log and use same std. normal functions
• norm.s.dist(z,cumulative) – returns probability
• norm.s.inv(p) – returns the value of the std normal
variate, z
Continuous Data
• Normal Distribution
- infinity to + infinity
• Lognormal Distribution
0 to + infinity
Beta Distribution
1
( x  L)a 1 (U  x) b 1
, L  x U
• Bounded: f X ( x) 
a  b 1
B(a , b )
(U  L)
where B(a,b) is the “beta function”
CDF
PDF
f X (x)
FX (x)
X
Images: http://en.wikipedia.org/wiki/Beta_distribution
X
Truncated Normal
1
• PDF:
where
and
f X ( x) 
( )
( )
s
(
(
U m
s
xm
s
)  (
)
Lm
s
, L  x U
)
is the PDF of the normal curve,
is the CDF of the normal curve
• Applicable for tolerance
ranges on dimensions
Uniform Distribution
• PDF:
1
f X ( x) 
, L  x U
U L
PDF
1
U L
L
• Applicable for tolerance ranges
on dimensions, temporal
variations
U
CDF
1.0
L
U
Mean and COV
• Expressions for E(X) and COV(X) for
different distributions