Stochastic Processes · 免费预览
多步转移概率
从状态 i 经过 m 步到达状态 j 的概率，是转移矩阵的 m 次幂的第 (i, j) 个元素，因为沿一条路径的概率相乘，而在各中间停靠点处则相加。
1. Two-stage questions
Two-stage questions
2. Matrix powers
Matrix powers
3. Evolving a distribution
Evolving a distribution
4. Powers of a four-state chain
Powers of a four-state chain
5. The one-one entry of the square
For the transition matrix with first row 0.5 and 0.5, second row 0.3 and 0.7, compute the one-one entry of p squared, that is the probability of going from state 1 back to state 1 in two steps.
6. The one-two entry of the square
For the same matrix, first row 0.5 and 0.5, second row 0.3 and 0.7, what is the probability of going from state 1 to state 2 in exactly two steps?
7. Why matrix multiplication
Why is the m-step transition probability the i, j entry of the mth power of the transition matrix?
8. Evolving a random start
A two-state chain has rows (0.5, 0.5) and (0.3, 0.7), and the initial distribution is q = (0.3, 0.7). What is the probability that X_1 is state 1?
9. Two steps in the ruin chain
In the gambler's ruin chain on states 0 to 4 with win probability 0.4, the fortune is 1. What is the probability of being at fortune 3 exactly two plays later?
10. Two more hits after two hits
In the two-stage basketball chain on states HH, HM, MH and MM, the current state is HH. What is the probability that the state is MH two shots later?
11. The two-two entry of the square
For the transition matrix with first row 0.5 and 0.5, second row 0.3 and 0.7, compute the probability of going from state 2 back to state 2 in two steps and write the number as a decimal.
12. Rain two days after sun
In the weather chain with p of rain given rain 0.6, rain given sun 0.2, sun given rain 0.4 and sun given sun 0.8, compute the probability that it rains two days after a sunny day and write the number as a decimal.
13. Return to HH after two shots
In the two-stage basketball chain, the current state is HH. Compute the probability that the state is HH again two shots later and write the answer as a decimal or a fraction.