1. Probability distributions

2. 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. : Wind loading and structural response
Lecture 3 Dr. J.D. Holmes, Reeding Univeristy

3. 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) x
fX(x) x 

4. 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) x
Pr(a<x<b)
x = a b
i.e.
Since all values of X must fall between - and +  :
i.e. total area under the graph of fX(x) is equal to 1

5. 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) x
x = a
Fx(a)
Area to the left of the x = a line is : FX(a)
This is the probability that X is less than a

6. 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) 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

7. Probability distributions
Moments of a distribution :
Mean value
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)

8. Probability distributions
Moments of a distribution :
Variance
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

9. Probability distributions
Moments of a distribution :
skewness
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` x
fx(x)
positive sx
negative sx
A distribution that is symmetrical about the mean value has zero skewness

10. Probability distributions
Gaussian (normal) distribution :
p.d.f.
fX(x) x
0.1
0.2
0.3
0.4 -4 -3 -2 -1 1 2 3 4
allows all values of x : -<x< +
bell-shaped distribution, zero skewness

11. Probability distributions
Gaussian (normal) distribution :
c.d.f. 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) =
Used for turbulent velocity fluctuations about the mean wind speed, dynamic structural response, but not for pressure fluctuations or scalar wind speed

12. Probability distributions
Lognormal distribution :
p.d.f.
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)

13. Probability distributions
Weibull distribution :
p.d.f. fX(x) =
complementary c.d.f. FX(x) =
c.d.f. FX(x) =
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

14. Probability distributions
Weibull distribution :
k=3
k=2
k=1 x
0.0
0.2
0.4
0.6
0.8
1.0
1.2 1 2 3 4
fx(x)
Special cases : k=1 Exponential distribution
k=2 Rayleigh distribution

15. 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) =
 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

16. 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

17. Probability distributions
Generalized Extreme Value distribution (G.E.V.) :
c.d.f FY(y) =
k is the shape factor; a is the scale factor; u is the location parameter
Special cases : Type I (k=0) Gumbel exp[exp(-(y-u)/a)]
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

18. Probability distributions
Generalized Extreme Value distribution (G.E.V.) : -6 -4 -2 2 4 6 8 -3 -1 1 3
Reduced variate : -ln[-ln(FY(y)]
(y-u)/a
Type I k = 0
Type III k = +0.2
Type II k = -0.2
(In this way of plotting, Type I appears as a straight line)
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)

19. Probability distributions
Generalized Pareto distribution (G.P.D.) :
c.d.f FX(x) =
k is the shape factor  is the scale factor
p.d.f. fX(x) =
k = 0 or k<0 : < X < 
k>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

20. Probability distributions
Generalized Pareto distribution :
fx(x)
x/
k=+0.5
k=-0.5
G.P.D. can be used with Poisson distribution of storm occurrences to predict extreme winds from storms of a particular type