Stochastic Processes · 免费预览
马尔可夫链的定义与示例
马尔可夫链是一列随机状态，其中仅凭当前状态就能确定下一步的转移；而它的转移矩阵——一个各行元素之和为一的非负元素表——就是完整的规则。
1. A game with no memory of the path
A game with no memory of the path
2. The transition matrix is the rulebook
The transition matrix is the rulebook
3. Chains from every walk of life
Chains from every walk of life
4. When one step of memory is not enough
When one step of memory is not enough
5. What the Markov property says
Which statement is the Markov property for a chain with transition probability p, in words?
6. One conditional probability in the ruin chain
In the gambler's ruin chain with win probability 0.4, the fortune is currently 2, and the previous fortunes were 1 then 2 then 1. What is the probability the next fortune is 3?
7. Reading a transition matrix by rows
Which condition must every row of a transition matrix satisfy, and why?
8. Reading the weather table
The weather chain has states 1 for rainy and 2 for sunny, with p(1,1) equal to 0.6, p(1,2) equal to 0.4, p(2,1) equal to 0.2 and p(2,2) equal to 0.8. Today is rainy. What is the probability that tomorrow is sunny?
9. The two-stage trick
A player's hitting probability depends on the last two shots. How is the process converted into a Markov chain?
10. A row from the brand chain
In the brand preference chain, customers satisfied with brand 1 repurchase it with probability 0.8, and otherwise pick one of the other two brands at random. What is the first row of the transition matrix?
11. A step down in the ruin chain
In the gambler's ruin chain with win probability 0.4, the current fortune is 2. Compute the probability that the next fortune is 1 and write the number as a decimal.
12. Ehrenfest draws from the other urn
In the Ehrenfest chain with 4 balls, the left urn currently holds 2. Compute the probability that the next step moves a ball into the left urn, and write the number as a decimal or fraction.
13. Two misses then a miss
In the two-stage basketball chain with states HH, HM, MH and MM, the current state is MM. Compute the probability that the next state is again MM and write the answer as a decimal or a fraction.