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The main points i the stochastic hydrology are listed below:Extreme Value Type, Gumbel’s Extreme Value Distribution, Maximum Values, Double Exponential Distribution, Annual Peak Flood, Weibull Distribution, Mean and Variance, Parent Distribution, Sample Moment Values
Typology: Study notes
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{ }
( ) exp exp
f x x x
x
m m
Y = (X – β)/ α → transformation
{ }
( ) { }
( ) { }
(maximum)
(minimum)
σ
ˆ
β μ σ
μ σ
=
= +
(maximum)
(minimum)
(Double Exponential
Distribution)
The annual peak flood of a stream exceeds 2000m
3
/s
with a probability of 0.02 and exceeds 2250m
3
/s with a
probability of 0.
2500m
3
/s
Solution:
The parameters α and β are obtained from the given data
as follows
i.e., = 0.
4
y
e
−
−
y
e
−
−
Solving (1) and (2),
α = 358 and β = 603
Now P[X > 2500] = 1 – P[X < 2500]
= 1 – exp{-exp(-y)}
y = (x – β)/ α
P[X > 2500] = 1 – exp{-exp(-5.299)}
Example-1 (contd.)
6
Extreme Value Type-III Distribution
μ = E[X] = β Γ(1+1/α)
σ
2
= Var(X) = β
2
{Γ(1+2/α) – Γ
2
(1+1/α)}
7
( )
{ }
1
α
α α
α β β α β
− −
( ) { }
α
= − − β ≥ α β >
Weibull Distribution
α
→ transformation
distribution are
μ = E[X] = ε + (β - ε) Γ(1+1/α)
σ
2
= Var(X) = (β- ε)
2
{Γ(1+2/α) – Γ
2
(1+1/α)}
9
Weibull Distribution
10
α → Shape parameter
β → Scale parameter
α=
β=
x
f(x)
α=
β=
α=
β=
α=
β=
α=
β=
Example-2 (contd.)
12
σ
2
= Var(X) = β
2
{Γ(1+2/α) – Γ
2
(1+1/α)}
2
2 2
2
2 2
2 2 2
β
α α
μ
α
μ = β Γ(1+1/α)
μ /β = Γ(1+1/α)
Substituting the
sample moment
values
Example-2 (contd.)
13
( )
{ }
( )
{ }
α
Parameter Estimation
f(x; θ
1
; θ
2
………… θ
m
) or F(x; θ
1
; θ
2
………… θ
m
1
, x
2
, …… x
n
is available
: Estimate of θ
i
: a function of the sample
sample.
§ Unbiasedness
§ Consistency
15
i
i
Parameter Estimation
Unbiased estimate:
E( ) = θ.
individual is equal to θ or even close to θ
equal to θ.
16
Parameter Estimation
Methods of estimating parameters from samples of data:
18
Method of matching points
parameters
parameters
19
Method of Moments (MoM)
the parameters
sample estimates of the first m moments
unknown parameters of the distribution.
21
Example-
Obtain the parameter λ using method of moments for
the pdf
The first moment is
22
x
λ
−
0
x
λ
∞
−
∫
0 0
2
0
0
x x
x x
x
xe e
dx
xe e
e x
λ λ
λ λ
λ
∞
∞ − −
∞
− −
∞
−