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 â€¢? Markov chains â€“? Transition probabilities â€“? Transition probability matrix (TPM) â€“? 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 â€¢? Markov chains â€“? Transition probabilities â€“? Transition probability matrix (TPM) â€“? Steady state Markov chains â€¢? 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 â€¢? 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 â€¢? Markov chains â€“? Transition probabilities â€“? Transition probability matrix (TPM) â€“? Steady state Markov chains â€¢? 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 â€¢? 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 â€“ 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 â€¢? Markov chains â€“? Transition probabilities â€“? Transition probability matrix (TPM) â€“? Steady state Markov chains â€¢? 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 â€¢? 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 â€“ 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 â€“ 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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