Probability Distribution in research methodology

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

• 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


• 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
• i.e. 
 dx
x
f
b
x
a
b
a
x



 )
(
Pr
• Since all values of X must fall between - and +  : 1
)
( 




dx
x
fx
• i.e. total area under the graph of fX(x) is equal to 1
fX(x)
x
Pr(a<x<b)
x = a b

• 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

• 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

• Moments of a distribution :
• Mean value
fX(x)
x

 











N
i
i
x x
N
dx
x
xf
X
1
1
)
(
• 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)
x =X

• Variance    

 













N
i
i
x
x X
x
N
dx
x
f
X
x
1
2
2
2 1
)
(

• The variance, X
2
, is the second moment of the probability distribution
about the mean value
fX(x)
x =X x
• It is equivalent to the second moment of area of a cross section about the
centroid
X
• The standard deviation, X, is the square root of the variance

• skewness    

 























N
i
i
X
x
X
x X
x
N
dx
x
f
X
x
s
1
3
3
3
3
1
)
(
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`
• A distribution that is symmetrical about the mean value has zero skewness
x
fx(x) positive sx
negative sx

• Gaussian (normal) distribution :
• p.d.f.
 







 

 2
x
2
x
x
2σ
X
x
exp
σ
2π
1
(x)
f
fX(x)
x
0
0.1
0.2
0.3
0.4
-4 -3 -2 -1 0 1 2 3 4
allows all values of x : -<x< +
bell-shaped distribution, zero skewness

• 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) =







 
X
X
x

dz
z
u

 






 






2
exp
2
1 2

Used for turbulent velocity fluctuations about the mean wind speed,
dynamic structural response, but not for pressure fluctuations or scalar
wind speed

• 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)



























 2
2
x
2σ
m
x
log
exp
x
σ
2π
1
(x)
f
e

• Weibull distribution :
p.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.






















  k
k
k
c
x
c
kx
exp
1
















k
c
x
exp
1
Weibull distribution widely used for wind speeds, and sometimes for pressure
coefficients
complementary
c.d.f. FX(x) =















k
c
x
exp

• Weibull distribution :
Special cases : k=1 Exponential distribution
k=2 Rayleigh distribution
k=3
k=2
k=1
x
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 1 2 3 4
fx(x)

• 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
Probability function : pX(x) =
number of exceedences of a defined pressure level on a building
 
!
exp
x
x 

  
!
exp
)
(
x
T
T 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

• 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
c.d.f of Y : FY(y) = FX1(y). FX2(y). ……….FXn(y)
Let Y be the maximum of n independent random variables, X1, X2, …….Xn
Special case - all Xi have the same c.d.f : FY(y) = [FX1(y)]n

• 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 III (k>0) ‘Reverse Weibull’
Type II (k<0) Frechet















 


k
a
u
y
k
/
1
)
(
1
exp
G.E.V (or Types I, II, III separately) - used for extreme wind speeds and
pressure coefficients

• Generalized Extreme Value distribution (G.E.V.) :
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)
-6
-4
-2
0
2
4
6
8
-3 -2 -1 0 1 2 3 4
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)

• 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 : 0 < X< (/k) i.e. upper limit
k = 0 or k<0 : 0 < X < 
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
k
1
σ
kx
1 












1
k
1
σ
kx
1
σ
1



























• Generalized Pareto distribution :
0.0
0.5
1.0
0 1 2 3 4
fx(x)
x/
k=+0.5
k=-0.5
0
G.P.D. can be used with Poisson distribution of storm occurrences to
predict extreme winds from storms of a particular type

Probability Distribution in research methodology

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