Stochastic Processes · 免费预览
计算转移概率
柯尔莫哥洛夫后向方程和前向方程通过速率矩阵 Q 将跳跃速率转化为转移概率，其解为矩阵指数 e^{Qt}，而像二态链和尤尔过程这样的小型系统可以显式求解。
1. From Chapman-Kolmogorov to a differential equation
From Chapman-Kolmogorov to a differential equation
2. The matrix exponential
The matrix exponential
3. The two-state chain solved by hand
The two-state chain solved by hand
4. The Yule process grows geometrically
The Yule process grows geometrically
5. Reading the Q matrix
How are the diagonal entries of the rate matrix Q defined?
6. Where the backward equation comes from
What reasoning produces Kolmogorov's backward equation?
7. The two solutions agree
Why is it remarkable that p sub t Q equals Q p sub t, and why is it true?
8. Reading the two-state solution
For flip rates alpha equal 1 and beta equal 3 per hour, what is the probability of finding state 1 after a long stretch of time?
9. How fast the start is forgotten
In the two-state chain, what sets the speed of convergence to equilibrium?
10. The Yule process law
Starting from one particle in a Yule process with birth rate beta, what is the distribution of the population at time t?
11. A machine that breaks and gets fixed
A machine flips from working to broken at rate alpha equal 1 per day and from broken to working at rate beta equal 3 per day. Compute the long-run probability that it is working, beta over alpha plus beta, and write the answer as a fraction or decimal.
12. The fading memory of the start
With the same flip rates 1 and 3 per hour, the deviation of p sub t of 1 comma 1 from its limit is one quarter times e to the minus 4 t. Compute this deviation at t equal to one half hour, and write the answer as a decimal.
13. A founding cell line after two hours
A Yule process has birth rate beta equal one half per hour. Starting from a single cell, compute the expected number of cells at time 2 hours, e to the beta t, and write the answer as a decimal.