Probability Distribution in research methodology

Probability distribution

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

5. Probability distributions
•Cumulative distribution function :
•The cumulative distribution function (c.d.f.) is the integral between -
and x of f
X(x)
•Denoted by F
X
(x)
f
X
(x)
x
x = a
F
x
(a)
•Area to the left of the x = a line is : F
X(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 f
X(x)
•Denoted by G
X
(x) and equal to : 1- F
X
(x)
f
X
(x)
x
G
x
(b)
x = b
•Area to the right of the x = b line is : G
X(b)
This is the probability that X is greater than b

7. Probability distributions
•Moments of a distribution :
•Mean value
f
X
(x)
x











N
i
ix x
N
dxxxfX
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 f
X(x)
x =X

8. Probability distributions
•Moments of a distribution :
•Variance   










N
i
ixx Xx
N
dxxfXx
1
222 1
)(
•The variance, 
X
2
, is the second moment of the probability distribution
about the mean value
f
X
(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

9. Probability distributions
•Moments of a distribution :
•skewness   






















N
i
i
X
x
X
x Xx
N
dxxfXxs
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
f
x
(x)
positive s
x
negative s
x

10. Probability distributions
•Gaussian (normal) distribution :
•p.d.f.










2
x
2
x
x
2σ
Xx
exp
σ2π
1
(x)f
f
X(x)
x

0
0.1
0.2
0.3
0.4
-4-3-2-10 1 2 3 4
allows all values of x : -<x< +
bell-shaped distribution, zero skewness

11. Probability distributions
•Gaussian (normal) distribution :
•c.d.f. F
X(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
Xx

dz
zu

 













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

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 log
ex)
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 m
2
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
xσ2π
1
(x)f
e

13. Probability distributions
•Weibull distribution :
p.d.f. f
X(x) =
c.d.f. F
X
(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
exp1
Weibull distribution widely used for wind speeds, and sometimes for pressure
coefficients
complementary
c.d.f. F
X(x) =















k
c
x
exp

14. Probability distributions
•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
f
x(x)

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
Probability function : p
X(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

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
c.d.f of Y : F
Y(y) = F
X1(y). F
X2(y). ……….F
Xn(y)
Let Y be the maximum of n independent random variables, X
1
, X
2
, …….X
n
Special case - all X
i
have the same c.d.f : F
Y
(y) = [F
X1
(y)]
n

17. Probability distributions
•Generalized Extreme Value distribution (G.E.V.) :
c.d.f. F
Y(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
uyk
/1
)(
1exp
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.) :
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(F
Y
(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)

19. Probability distributions
•Generalized Pareto distribution (G.P.D.) :
c.d.f. F
X(x) =
k is the shape factor  is the scale factor
p.d.f. f
X
(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

























20. Probability distributions
•Generalized Pareto distribution :
0.0
0.5
1.0
0 1 2 3 4
f
x(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