Stochastic Processes · 免费预览
退出时间
到达目标点的期望时间满足首步递推关系，其中要为已耗费的一步加上 1；该方程组可通过 I-r 的逆矩阵求解。公平赌徒破产问题的持续时间为 x(N-x)，这便是示例公式。
1. Timing the exit
Timing the exit
2. Counting visits to measure time
Counting visits to measure time
3. Why two tails take longer than a head-tail
Why two tails take longer than a head-tail
4. How long fair games last
How long fair games last
5. Why the plus one appears
Why does the expected exit time g satisfy g of x equal to one plus the sum of p of x, y times g of y, rather than the same sum without the one?
6. Boundary values for times
What values does the expected exit time g take on the target set A, and why?
7. Time as accumulated visits
Why is the expected exit time computable as I minus r inverse applied to the all-ones vector?
8. Two tails outlast a head-tail
Both patterns HT and TT have probability one quarter on any two fixed flips, yet the expected wait for TT is 6 and for HT is 4. What explains the gap?
9. The fair ruin parabola
For fair gambler's ruin on zero to N, which properties does the expected duration E of x tau equal x times N minus x have?
10. The unfair-game shortcut
For a gambler losing at rate q minus p per play, starting at x, why is the expected time to go broke roughly x over q minus p?
11. The pennies game clock
In matching pennies with 25 total coins, Bob starts with 15. Using the fair ruin duration formula, compute the expected number of flips until someone wins all the pennies.
12. A sophomore's shelf life
At the two-year college, a sophomore repeats with probability 0.2, graduates with 0.7 and drops out with 0.1. Compute the expected years until a sophomore leaves, and write the answer as a fraction or decimal.
13. Waiting for two tails
Compute the expected number of fair coin flips needed to see tails twice in a row, and write the number.