Stochastic Processes · 免费预览
平稳分布
平稳分布就像一行沙粒，转移矩阵可以将其重新排列而不改变它；它满足 pi 乘以 p 等于 pi，并且每个有限链至少有一个平稳分布。
1. Sand that does not move
Sand that does not move
2. The two-state formula
The two-state formula
3. Three states and elimination
Three states and elimination
4. Doubly stochastic chains
Doubly stochastic chains
5. What stationarity means
What does it mean to say that pi is a stationary distribution for a chain with transition matrix p?
6. The weather chain's equilibrium
Using the two-state formula with a equal to 0.4 and b equal to 0.2, what is the stationary distribution of the weather chain?
7. Balance across the gap
For a two-state chain with jump rates a from state 1 to 2 and b from state 2 to 1, what is the balance condition behind the stationary formula?
8. The elimination procedure
When solving pi p equals pi for a three-state chain, why does one replace one of the balance equations with the normalizing equation?
9. Doubly stochastic shortcut
A chain on five states has a transition matrix whose columns each sum to one. What is its stationary distribution?
10. Market shares at equilibrium
The brand preference chain has rows (0.8, 0.1, 0.1), (0.2, 0.6, 0.2) and (0.3, 0.3, 0.4). What is the stationary share of brand 1?
11. The sunny share
For the weather chain with a equal to 0.4 and b equal to 0.2, compute the stationary probability of the sunny state and write the answer as a fraction or a decimal.
12. Uniform weight on five states
A doubly stochastic chain runs on five states. Compute the stationary probability of any single state and write the answer as a decimal or a fraction.
13. The upper class at equilibrium
For the social mobility chain with rows (0.7, 0.2, 0.1), (0.3, 0.5, 0.2) and (0.2, 0.4, 0.4), compute the stationary probability of the upper class, state 3, and write the answer as a fraction.