Stochastic Processes · 免费预览
细致平衡与可逆性
细致平衡比平稳性要求更多：沿每一条边的沙流量都相互抵消；而对于生灭链、「Ehrenfest」罐模型以及图上的随机游走，它自动成立。因此，细致平衡既是一种计算上的捷径，也是可逆性的定义。
1. Cancellation along every edge
Cancellation along every edge
2. Birth-death chains balance edge by edge
Birth-death chains balance edge by edge
3. The Ehrenfest urn and its binomial equilibrium
The Ehrenfest urn and its binomial equilibrium
4. Random walks on graphs and reversibility
Random walks on graphs and reversibility
5. What detailed balance adds
How does detailed balance strengthen the stationary equation pi p equals pi?
6. The repairman's equilibrium
In the three machines one repairman chain, the detailed-balance recursion gives relative weights 1, 10, 45 and 60 on states 0 through 3. What is the stationary probability that all three machines work?
7. Ehrenfest's binomial pattern
What is the stationary distribution of the Ehrenfest chain with three balls, and what is the underlying reason?
8. Stationary weights on a graph
For a random walk on a graph, how does the stationary probability of a vertex depend on the graph?
9. Stationary without detailed balance
A certain three-state chain has the uniform vector stationary but has an edge with strictly positive probability in one direction and zero in the other. What can be concluded?
10. The reversed chain's matrix
A stationary chain with distribution pi and matrix p is run backward in time. What is the transition probability of the reversed chain from i to j?
11. Middle weights in the urn
For the Ehrenfest chain with three balls, compute the stationary probability of state 1, exactly one ball in the left urn, and write the answer as a fraction or a decimal.
12. A crossing flow in the urn
In the Ehrenfest chain with three balls, compute the stationary flow from state 1 to state 2, that is pi of 1 times p of 1, 2, and write the answer as a fraction or a decimal.
13. A knight on the corner
A knight performs a random walk on a chessboard, where the degrees of all 64 squares sum to 336. Compute the stationary probability of standing on a given corner square, which has degree 2, and write the answer as a fraction.