Stochastic Hydrology, Civil Engineering

# Stochastic Hydrology, Civil Engineering - Notes - Civil Engineering (CE)

``` Page 1

STOCHASTIC HYDROLOGY
Lecture -23
Course Instructor :  Prof. P. P. MUJUMDAR
Department of Civil Engg., IISc.

INDIAN
INSTITUTE
OF
SCIENCE

Page 2

STOCHASTIC HYDROLOGY
Lecture -23
Course Instructor :  Prof. P. P. MUJUMDAR
Department of Civil Engg., IISc.

INDIAN
INSTITUTE
OF
SCIENCE

2

Summary
of
the
previous
lecture

•? Markov chains
–? Transition probabilities
–? Transition probability matrix (TPM)

Page 3

STOCHASTIC HYDROLOGY
Lecture -23
Course Instructor :  Prof. P. P. MUJUMDAR
Department of Civil Engg., IISc.

INDIAN
INSTITUTE
OF
SCIENCE

2

Summary
of
the
previous
lecture

•? Markov chains
–? Transition probabilities
–? Transition probability matrix (TPM)

•? stochastic process with the property that value of
process X
t
at time t depends on its value at time t-1
and not on the sequence of other values

3

Markov Chains
1
t
tj t i ij
PX a X a P
-
????
===
????
( ) ( )
0 n
n
pp P = ×
[ ] [ ]
12 0 1
,,.....
tt t tt
PX X X X PX X
-- -
=
ppP = ×
Page 4

STOCHASTIC HYDROLOGY
Lecture -23
Course Instructor :  Prof. P. P. MUJUMDAR
Department of Civil Engg., IISc.

INDIAN
INSTITUTE
OF
SCIENCE

2

Summary
of
the
previous
lecture

•? Markov chains
–? Transition probabilities
–? Transition probability matrix (TPM)

•? stochastic process with the property that value of
process X
t
at time t depends on its value at time t-1
and not on the sequence of other values

3

Markov Chains
1
t
tj t i ij
PX a X a P
-
????
===
????
( ) ( )
0 n
n
pp P = ×
[ ] [ ]
12 0 1
,,.....
tt t tt
PX X X X PX X
-- -
=
ppP = ×
Example – 1
4

Consider the TPM for a 2-state first order homogeneous
Markov chain as

State 1 is a non-rainy day and state 2 is a rainy day
Obtain the
1.? probability that day 1 is a non-rainy day given that day 0 is
a rainy day
2.? probability that day 2 is a rainy day given that day 0 is a
non-rainy day
3.? probability that day 100 is a rainy day given that day 0 is a
non-rainy day
0.7 0.3
0.4 0.6
TPM
????
=
????
????
Page 5

STOCHASTIC HYDROLOGY
Lecture -23
Course Instructor :  Prof. P. P. MUJUMDAR
Department of Civil Engg., IISc.

INDIAN
INSTITUTE
OF
SCIENCE

2

Summary
of
the
previous
lecture

•? Markov chains
–? Transition probabilities
–? Transition probability matrix (TPM)

•? stochastic process with the property that value of
process X
t
at time t depends on its value at time t-1
and not on the sequence of other values

3

Markov Chains
1
t
tj t i ij
PX a X a P
-
????
===
????
( ) ( )
0 n
n
pp P = ×
[ ] [ ]
12 0 1
,,.....
tt t tt
PX X X X PX X
-- -
=
ppP = ×
Example – 1
4

Consider the TPM for a 2-state first order homogeneous
Markov chain as

State 1 is a non-rainy day and state 2 is a rainy day
Obtain the
1.? probability that day 1 is a non-rainy day given that day 0 is
a rainy day
2.? probability that day 2 is a rainy day given that day 0 is a
non-rainy day
3.? probability that day 100 is a rainy day given that day 0 is a
non-rainy day
0.7 0.3
0.4 0.6
TPM
????
=
????
????
Example – 1 (contd.)
5

1.? probability that day 1 is a non-rainy day given that day
0 is a rainy day
The probability is 0.4

2.? probability that day 2 is a rainy day given that day 0 is
a non-rainy day
p
(1)
, in this case is [0.7 0.3] because it is given that
day 0 is a non-rainy day.
0.7 0.3
0.4 0.6
TPM
????
=
????
????
No rain

rain

No rain

rain

( ) ( )
21
pp P = ×
```

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