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WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
WFM-6204: Hydrologic Statistics
Lecture-3: Probabilistic analysis: (Part-2)
Akm Saiful Islam
Institute of Water and Flood Management (IWFM)
Bangladesh University of Engineering and Technology (BUET)
December, 2006
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Probability Distributions and Their
Applications
 Continuous
Distributions
Normal distribution
 Lognormal distribution
 Gamma distribution
 Pearson Type III distribution
 Gumbel’s Extremal distribution

WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Normal Distribution
The probability that X is less than or equal to x when X can
be evaluated N ( ,  2 ) from
x
prob( X  x)  px ( x)   (2 )

2 1/ 2
e
( t  )2 / 2 2
dt
(4.9)
The parameters  (mean) and  2 (variance) are
denoted as location and scale parameters, respectively.
The normal distribution is a bell-shaped, continuous and
symmetrical distribution (the coefficient of skew is zero).
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
 If  is held constant and
2
varied, the
distribution changes as in Figure 4.2.1.1.
Figure 4.2.1.1
Normal distributions with same mean and different variances
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam

2
If  is held constant and  varied, the distribution does
not change scale but docs change location as in Figure
4.2.1.2. A common notation for indicating that a random
variable is normally distributed with mean  and
variance  2 is N ( ,  2 )
Figure 4.2.1.2
Normal distributions with same variance and different means
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam

If a random variable is N ( ,  2 ) and Y= a + bX
2 2
, the distribution of Y can be shown to be N (a  b, b.  )
This can be proven using the method of derived
i  1,2, n
distributions. Furthermore, if for X , are
independently and normally distributed with
mean  i and variance  , then Y  a  b X  b X    b X
is normally distributed with
i
2
i
Y  a  i 1bi i
n
  i 1bi2 i2
2
Y

n
1
1
2
2
(4.7) and
(4.8)
Any linear function of independent normal
random variables is also a normal random
variable.
n
n
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Standard normal distribution

The probability that X is less than or equal
to x when X is N ( ,  ) can be evaluated
from
2
x
prob( X  x)  px ( x)   (2 )

2 1/ 2
e
( t  )2 / 2 2
dt
(4.9)
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam

The equation (4.9) cannot be evaluated
analytically so that approximate methods of
integration arc required. If a tabulation of the
integral was made, a separate table would be
required for each value of  and  . By using the
liner transformation Z  ( X   ) / , the random variable
Z will be N(0,1). The random variable Z is said to
be standardized (has   0 and   1 ) and N(0,1) is
said to be the standard normal distribution. The
standard normal distribution is given by
   z   (4.10)
p ( z)  (2 ) e
and the cumulative standard normal is given by
z
prob( Z  z)  PZ ( z)   (2 ) 1 / 2 e t / 2 dt (4.11)
2
2
1 / 2
z2 / 2
Z
2

WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Figure 4.2.1.3
Standard normal distribution
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam





Figure 4.2.1.3 shows the standard normal
distribution which along with the transformation
Z  ( X   ) /  contains all of the information shown
in Figures 4.1 and 4.2.
Both pZ (Z )
and PZ (z) are widely
tabulated.
Most tables utilize the symmetry of the normal
distribution so that only positive values of Z are
shown.
Tables of PZ (z) may show prob( Z  z ) or
prob (0  Z  z )
Care must be exercised when using normal
probability tables to see what values are
tabulated.
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam

Example-1: As an example of using tables of the
normal distribution consider a sample drawn from
a N(15,25). What is the prob(15.6≤ X≤ 20.4)?
[Hann]

Solution:
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam

Example-2: What is the prob(10.5≤ X≤ 20.4) if X
distributed N(15,25)? [Hann]

Solution:
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam

Example-3: Assume the following data follows
a normal distribution. Find the rain depth that
would have a recurrence interval of 100 years.

Year
2000
1999
1998
1997
1996
…..






Annual Rainfall (in)
43
44
38
31
47
…..
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Solution:
Mean = 41.5, St. Dev = 6.7 in (given)
x= Mean + Std.Dev * z
x = 41.5 + z(6.7)
P(z) = 1/T = 1/100 = 0.01
F(z) = 0.5 – P(z) = 0.49
From Interpolation using Tables E.4
Z = 2.236
X = 41.5 + (2.326 x 6.7) = 57.1 in
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Properties of common distributions
Bi-Nominal Distribution
PDF
P ( x) 
Range
0 xn
Mean
np
Variance
np (1  p )
n!
p x (1  p ) n  x
x! (n  x)!
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Properties of common distributions
Poisson Distribution
PDF
Range
P( x) 
 x e 
x!
0  x  ...
Mean

Variance

WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Properties of common distributions
Normal Distribution
PDF
f ( x) 
1
 2
Range
  x 
Mean

Variance
2
e
 ( x   ) 2 / 2 2
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Properties of common distributions
Log-Normal Distribution
( y = ln x)
PDF
1
( y   ) 2 / 2 2
f ( y) 
e
x 2
Range
  y 
Mean

Variance
y 2
y
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Properties of common distributions
Gamma Distribution
PDF
Range
Mean
Variance
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Properties of common distributions
Gumbel Distribution
PDF
Range
Mean
Variance
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Properties of common distributions
Extreme Value Type-1 Distribution
PDF
Range
Mean
Variance
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Properties of common distributions
Log-Pearson III Distribution
PDF
Range
Mean
Variance
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Assignment-1

The total annual runoff from a small
drainage basin is determined to be
approximately normal with a mean of 14.0
inch and a variance of 9.0 inch2.
Determine the probability that the annual
runoff form the basin will be less than 11.0
inch in all three of next the three
consecutive years.