Download Probability distributions

Wind loading and structural response
Lecture 3 Dr. J.D. Holmes
Probability distributions
Probability distributions
• Topics :
• Concepts of probability density function (p.d.f.) and
cumulative distribution function (c.d.f.)
• Moments of distributions (mean, variance, skewness)
• Parent distributions
• Extreme value distributions
Ref. : Book - Appendix C
Probability distributions
• Probability density function :
• Limiting probability (x  0) that the value of a random variable X
lies between x and (x + x)
• Denoted by fX(x)

fX(x)
x
x
Probability distributions
• Probability density function :
• Probability that X lies between the values a and b is the area under the
graph of fX(x) defined by x=a and x=b
fX(x)
Pr(a<x<b)
x=a
• i.e.
b
x
Pra  x  b   f x ( x)dx
b
a
• Since all values of X must fall between - and +  :
• i.e. total area under the graph of fX(x) is equal to 1



f x ( x)dx  1
Probability distributions
• Cumulative distribution function :
• The cumulative distribution function (c.d.f.) is the integral between -
and x of fX(x)
• Denoted by FX(x)
fX(x)
Fx(a)
x= a
• Area to the left of the x = a line is : FX(a)
This is the probability that X is less than a
x
Probability distributions
• Complementary cumulative distribution function :
• The complementary cumulative distribution function is the integral
between x and + of fX(x)
• Denoted by GX(x) and equal to : 1- FX(x)
fX(x)
Gx(b)
x= b
• Area to the right of the x = b line is : GX(b)
This is the probability that X is greater than b
x
Probability distributions
• Moments of a distribution :
• Mean value
1N
X   xfx ( x)dx    xi

 N  i 1

fX(x)
x =X
x
• The mean value is the first moment of the probability distribution, i.e.
the x coordinate of the centroid of the graph of fX(x)
Probability distributions
• Moments of a distribution :
• Variance
x  
2


x  X 
2
2
1N
f x ( x)dx    xi  X 
 N  i 1
• The variance, X2, is the second moment of the probability distribution
about the mean value
• It is equivalent to the second moment of area of a cross section about
the centroid
• The standard deviation, X, is the square root of the variance
fX(x)
X
x =X
x
Probability distributions
• Moments of a distribution :
• skewness
 1  
 1 N
3
3
s x   3   x  X  f x ( x)dx   3  xi  X 

 X 
  X N  i 1
• Positive skewness indicates that the distribution has a long tail on the
positive side of the mean
• Negative skewness indicates that the distribution has a long tail on the
negative side of the mean`
fx(x)
positive sx
negative sx
x
• A distribution that is symmetrical about the mean value has zero
skewness
Probability distributions
• Gaussian (normal) distribution :
• p.d.f.
  x  X 2 
1
f x (x) 
exp 

2
2σ
2πσ x


x
0.4
fX(x)
0.3
0.2
0.1
0
-4
-3
-2
-1
0
1
allows all values of x : -<x< +
bell-shaped distribution, zero skewness
2
x
3
4
Probability distributions
• Gaussian (normal) distribution :
• c.d.f.
 x X 


 X 
FX(x) =  
( ) is the cumulative distribution function of a normally distributed
variable with mean of zero and unit standard deviation (tabulated in
textbooks on probability and statistics)
 (u) =
  z2 
 1  u
dz

 exp 

 2 
 2 
Used for turbulent velocity fluctuations about the mean wind speed,
dynamic structural response, but not for pressure fluctuations or scalar
wind speed
Probability distributions
• Lognormal distribution :
• p.d.f.
2
 
 x  
  log e   
1
 m  
f x (x) 
exp  


2σ 2
2πσ x




A random variable, X, whose natural logarithm has a normal
distribution, has a Lognormal distribution
(m,  are the mean and standard deviation of logex)
Since logarithms of negative values do not exist, X > 0
the mean value of X is equal to m exp (2/2)
the variance of X is equal to m2 exp(2) [exp(2) -1]
the skewness of X is equal to [exp(2) + 2][exp(2) - 1]1/2 (positive)
Used in structural reliability, and hurricane modeling (e.g. central pressure)
Probability distributions
• Weibull distribution :
  x k 
 kxk 1 
p.d.f. fX(x) =  k  exp    
  c  
 c 
  x k 
c.d.f. FX(x) = 1  exp    
  c  
complementary
  x k 
c.d.f. FX(x) = exp    
  c  
c = scale parameter (same units as X)
k= shape parameter (dimensionless)
X must be positive, but no upper limit.
Weibull distribution widely used for wind speeds, and sometimes for pressure
coefficients
Probability distributions
• Weibull distribution :
1.2
k=3
1.0
fx(x)
0.8
k=2
0.6
0.4
k=1
0.2
0.0
0
1
2
3
x
Special cases : k=1 Exponential distribution
k=2 Rayleigh distribution
4
Probability distributions
• Poisson distribution :
Previous distributions used for continuous random variables (X can take
any value over a defined range)
Poisson distribution applies to positive integer variables
Examples : number of hurricanes occurring in a defined area in a given time
number of exceedences of a defined pressure level on a building
Probability function : pX(x) = x
exp  T 
exp   
 (T ) x
x!
x!
 is the mean value of X. Standard deviation = 1/2
 is the mean rate of ocurrence per unit time. T is the reference time period
Probability distributions
• Extreme Value distributions :
Previous distributions used for all values of a random variables, X
- known as ‘parent distributions
In many cases in civil engineering we are interested in the largest
values, or extremes, of a population for design purposes
Examples : flood heights, wind speeds
Let Y be the maximum of n independent random variables, X1, X2, …….Xn
c.d.f of Y :
FY(y) = FX1(y). FX2(y). ……….FXn(y)
Special case - all Xi have the same c.d.f : FY(y) = [FX1(y)]n
Probability distributions
• Generalized Extreme Value distribution (G.E.V.) :
c.d.f.
1/ k

  k ( y  u)  

FY(y) = exp  1 


a

 
 

k is the shape factor; a is the scale factor; u is the location parameter
Special cases : Type I (k=0) Gumbel
Type II (k<0) Frechet
Type III (k>0) ‘Reverse Weibull’
G.E.V (or Types I, II, III separately) - used for extreme wind speeds and
pressure coefficients
Probability distributions
• Generalized Extreme Value distribution (G.E.V.) :
Type I k = 0
Type III k = +0.2
Type II k = -0.2
8
(In this way of
plotting, Type I
appears as a straight
line)
6
(y-u)/a
4
2
0
-3
-2
-1
-2
0
1
2
3
4
-4
-6
Reduced variate : -ln[-ln(FY(y)]
Type I, II :
Y is unlimited as c.d.f. reduces
Type III:
Y has an upper limit
(may be better for variables with an expected physical upper limit such as wind speeds)
Probability distributions
• Generalized Pareto distribution (G.P.D.) :
c.d.f.
FX(x) =
  kx 
1   σ 
  
1
k
 is the scale factor
k is the shape factor
p.d.f. fX(x) =
 1   kx 
 1   
 σ    σ 
k = 0 or k<0 :
k>0 :
1
  1
k
0<X<
0 < X< (/k)
i.e. upper limit
G.P.D. is appropriate distribution for independent observations of excesses
over defined thresholds
e.g. thunderstorm downburst of 70 knots. Excess over 40 knots is 30 knots
Probability distributions
• Generalized Pareto distribution :
1.0
fx(x)
k=+0.5
0
0.5
k=-0.5
0.0
0
1
2
x/
3
4
G.P.D. can be used with Poisson distribution of storm occurrences to
predict extreme winds from storms of a particular type
End of Lecture 3
John Holmes
225-405-3789 [email protected]