Civil Engineering (CE) > Stochastic Hydrology, Civil Engineering

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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) –? Steady state Markov chains 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) –? Steady state Markov chains •? 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 •? At steady state, 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) –? Steady state Markov chains •? 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 •? At steady state, 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) –? Steady state Markov chains •? 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 •? At steady state, 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 = ×Read More

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