Transcript Document
© 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 prob (
X
x
)
p x
(
x
)
x
( 2 2 ) 1 / 2
e
(
t
) 2 / 2 2
dt
(4.9) 2 The parameters (mean) and (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 2 If is held constant and 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 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 ( , 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 and Y= a + bX , the distribution of Y can be shown to be . 2 2 ) This can be proven using the method of derived distributions. Furthermore, if for , are
i
, 2 ,
n
independently and normally distributed with
i
i
2 is normally distributed with
Y
a
b
1
X
1
b
2
X
2
b n X n
Y
a
n i
1
b i
i
Y
2
n i
1
b i
2
i
2 (4.7) and (4.8) Any linear function of independent normal random variables is also a normal random variable.
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Standard normal distribution
The probability that X is less than or equal from prob (
X
x
)
p x
(
x
)
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 , the random variable Z will be N(0,1). The random variable Z is said to be standardized (has and ) and N(0,1) is said to be the standard normal distribution. The standard normal distribution is given by
p Z
(
z
) ( 2 ) 1 / 2
e
z
2 / 2
z
(4.10) and the cumulative standard normal is given by prob(
Z
z
)
P Z
(
z
)
z
( 2 ) 1 / 2
e
t
2 / 2
dt
(4.11)
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.
p Z
( tabulated.
Z
)
P Z
(
z
Most tables utilize the symmetry of the normal distribution so that only positive values of Z are shown. Tables of prob ( 0
Z P Z
(
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 Annual Rainfall (in) 43 44 38 1997 31 1996 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 Range Mean
P
(
x
)
x
!
(
n n
!
x
)!
p x
( 1
p
)
n
x
0
x
n np
Variance
np
( 1
p
)
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Properties of common distributions
Poisson Distribution
P
(
x
)
x e
x
!
Range Mean Variance 0
x
...
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Properties of common distributions
Normal Distribution
f
(
x
) 1 2
e
(
x
) 2 / 2 2 Range Mean Variance
x
2
WFM 6204: Hydrologic Statistics © Dr. Akm Saiful Islam
Properties of common distributions
Log-Normal Distribution ( y = ln x)
f
(
y
)
x
1 2
e
(
y
) 2 / 2 2 Range Mean Variance
y
y
y
2
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
inch 2 .
Determine the probability that the annual runoff form the basin will be less than 11.0
inch in all three consecutive years.
of next the three