Probability Distributions: Concepts, Moments, and Special Distributions, Slides of Environmental Law and Policy

Download Probability Distributions: Concepts, Moments, and Special Distributions and more Slides Environmental Law and Policy in PDF only on Docsity!

• 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 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.  a^^ x b  f xdx

b

 a x

Pr    ( )

• Since all values of X must fall between - and +  :  (^ ) ^1



 

fx xdx

• 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

• Moments of a distribution :
  • Mean value

fX(x)

x

  



 

N

i

x xi N

X xf xdx

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

• Moments of a distribution :
  • Variance   (^)    





N

i

x x xi X N

x X f x dx

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

• Gaussian (normal) distribution :
  • p.d.f.

 

2 x

2

x

x 2 σ

x X exp 2 π σ

f (x)

fX(x)

x

0

-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

u z

 (^)  

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

• 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 1 exp

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=

k=

k=

x

0 1 2 3 4

fx(x)

• 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, X 1 , X 2 , …….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

Type III (k>0) ‘Reverse Weibull’

Type II (k<0) Frechet

k

a

k y u

1 / ( ) exp 1

G.E.V (or Types I, II, III separately) - used for extreme wind speeds and

pressure coefficients

• 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

 

  



• Generalized Pareto distribution :

0 1 2 3 4

fx(x)

x/

k=+0.

k=-0.

0

G.P.D. can be used with Poisson distribution of storm occurrences to

predict extreme winds from storms of a particular type