Stochastic Processes · 免费预览
稀疏化、叠加与条件化
用独立的抛硬币结果为泊松到达事件着色，会将一个过程分裂为若干相互独立的泊松过程，其速率按各颜色的概率缩放；相互独立的过程可通过将速率相加而合并；而在给定到达次数的条件下，到达时刻会变成均匀次序统计量。
1. Splitting a stream by coin flips
Splitting a stream by coin flips
2. Two editors and missing typos
Two editors and missing typos
3. Merging streams and racing them
Merging streams and racing them
4. Given the count, the times are uniform
Given the count, the times are uniform
5. What thinning produces
A Poisson process of rate 10 per hour has each arrival independently labeled type A with probability 0.6 and type B with probability 0.4. What does the thinning theorem say?
6. The independence surprise
Why is the independence of thinned processes counterintuitive yet true?
7. Merging independent streams
Customers, emails and faxes arrive as independent Poisson processes with rates 5, 3 and 2 per hour. What is the combined arrival process?
8. The race trick
A red process of rate alpha races a green process of rate beta. Why is the probability of 6 red arrivals before 4 green ones computed from the first 9 combined arrivals?
9. Conditioning erases the rate
Given that a rate-lambda process had exactly n arrivals by time t, what is the joint distribution of the arrival times?
10. The binomial corollary
Given N of t equal to n, what is the distribution of the count N of s for s below t?
11. A mixed catch
Ellen fishes for 2.5 hours at Poisson rate 2 per hour, with 40 percent salmon and 60 percent trout. Compute the probability of exactly 1 salmon and exactly 2 trout and write the answer as a decimal.
12. A quiet salmon hour
For the same fishing setup, compute the rate of the thinned salmon process in arrivals per hour and write the answer as a decimal or fraction.
13. One in the first hour
Four customers arrived at a store during a three-hour window, a Poisson process at an unknown rate. Using the uniform conditioning law, compute the probability that exactly one of them arrived in the first hour and write the answer as a fraction or decimal.