Stochastic Processes · 免费预览
指数分布
速率为 lambda 的指数分布均值为 lambda 的倒数，忘记此前已经等待了多久，在按速率份额决定胜负的公平竞赛中发挥作用，且求和后归入 gamma 族。
1. A clock with a rate
A clock with a rate
2. Forgetting the wait so far
Forgetting the wait so far
3. Exponential races
Exponential races
4. Sums become gamma
Sums become gamma
5. Mean and variance of a rate
A device fails at rate lambda equal to 3 per year, its failure time exponential. What are the mean and variance of the failure time?
6. The restart property
You have already waited 10 minutes for a bus whose arrival time is exponential with mean 10 minutes. What is the probability you wait more than 10 additional minutes?
7. Deciding a race by rate share
S is exponential with rate lambda and T independent exponential with rate mu. What is the probability that S is smaller than T?
8. The fastest of many clocks
Five independent machines fail at rates 2, 3, 4, 5 and 6 per year. What is the distribution of the time until the first failure?
9. Winner and winning time
In an exponential race between several clocks, what is the relationship between the identity of the winner and the time at which it wins?
10. Sums of waiting times
Arrivals come one after another with independent exponential interarrival times of rate lambda. What is the distribution of the time of the third arrival?
11. A half-rate clock
A waiting time T is exponential with rate 0.5 per minute. Compute the expected waiting time E of T in minutes and write the number.
12. Who finishes first
Anne's manicure takes an exponential time with mean 20 minutes and Betty's haircut an independent exponential time with mean 30 minutes. Compute the probability Anne finishes first and write it as a fraction or decimal.
13. Both done at the salon
With Anne's exponential mean 20 minutes and Betty's independent exponential mean 30 minutes, the first completion comes at combined rate and then memorylessness prices the remainder. Compute the expected number of minutes until both are done and write the number.