Introduction to Probability · 免费预览
平稳分布、收敛性与可逆性
求解 sQ = s 所得的长期均衡向量、链向该向量的收敛、细致平衡与可逆性，以及 PageRank 的应用。
1. The stationary distribution
The stationary distribution
2. Convergence and when it holds
Convergence and when it holds
3. Detailed balance and reversibility
Detailed balance and reversibility
4. The stationarity equation
A row vector s is stationary for transition matrix Q when which equation holds?
5. Convergence's hypotheses
A finite chain is irreducible and aperiodic. What follows?
6. Two-state stationary
A chain stays at state 0 with probability 0.8 (so it leaves for state 1 with probability 0.2) and stays at 1 with probability 0.5 (leaving for 0 with 0.5). Balance requires 0.2 times s_0 to equal 0.5 times s_1, so s_1 = 0.4 s_0. With s_0 + s_1 = 1, what is s_0? Enter a fraction like 5/7.
7. Long-run availability
The server chain, up-stay 0.95, repair 0.6, has stationary (12/13, 1/13). What fraction of days is the server UP in the long run? Enter a decimal like 0.92.
8. Reversibility check
For a random walk on an undirected graph with 4 nodes, node A has degree 3 and node B has degree 1. In the stationary distribution proportional to degrees, the ratio P(A) over P(B) equals what? Enter an integer.
9. In your own words
In your own words, state the most important definition, theorem or method of this lesson and explain what it says.