Stochastic Processes · 免费预览
复合泊松过程
为每个泊松到达事件附加一个独立同分布的随机标记，会产生一个随机和；其均值为到达计数期望与标记均值之积，其方差则由标记的变异性和计数的变异性共同构成。
1. Arrivals that carry payloads
Arrivals that carry payloads
2. Averaging a random sum
Averaging a random sum
3. The Poisson bonus
The Poisson bonus
4. Accumulating over time
Accumulating over time
5. What makes a process compound
Which ingredients define a compound Poisson process?
6. Averaging over the count
What does Wald's identity say for a random sum S equal to Y one plus through Y sub N, with the marks independent of N?
7. The two sources of spread
In the variance formula for a random sum, what do the two terms E of N times var of Y and var of N times E of Y squared measure?
8. The Poisson simplification
When the count N is Poisson with mean lambda t, what does the variance formula for the compound sum reduce to?
9. Checking the formula at the edges
Which two special cases confirm that the random-sum variance formula has the right shape?
10. Reading the liquor store numbers
A store averages 81 customers a day, each spending a mean of 8 dollars with standard deviation 6 dollars. What are the mean and standard deviation of daily revenue?
11. A day of receipts
A liquor store averages 81 customers per day, each spending an exponential-free mean of 8 dollars with standard deviation 6 dollars. Compute the expected daily revenue in dollars and write the number.
12. How steady is the till
For the same store, use the Poisson special case to compute the standard deviation of daily revenue in dollars and write the number.
13. An hour of email
Emails arrive as a Poisson process with 90 expected in the next hour, each with mean size 4 kilobytes and standard deviation 3 kilobytes. Compute the expected total traffic in kilobytes and write the number.