Stochastic Processes · 免费预览
退出分布
在命中禁集之前经由目标集退出的概率满足首步递推关系，该递推可作为线性系统求解；对于公平博弈以及有偏赌徒破产问题，都有闭式解。
1. Where the chain ends up
Where the chain ends up
2. The linear system and its uniqueness
The linear system and its uniqueness
3. Fair games finish linear
Fair games finish linear
4. Biased ruin and the roulette lesson
Biased ruin and the roulette lesson
5. Boundary values of hitting probabilities
For the exit probability h of x defined as the chance of visiting the target A before the trap B, what values does h take on A and on B?
6. Where the first-step equation comes from
Why does the exit probability h satisfy h of x equal to the sum over y of p of x, y times h of y for x outside A and B?
7. The linear system behind the scenes
How are the first-step equations for h organized into a linear system?
8. The fair-game shortcut
In matching pennies, Bob starts with 15 of the 25 total pennies. Why is his win probability exactly 15 over 25?
9. Reading the biased ruin formula
In the biased gambler's ruin formula with theta equal to q over p, what happens to the win probability when p exceeds one half?
10. The lesson of the roulette table
A red-bettor starts with 50 dollars and aims for 100 on a wheel with 18 red, 18 black and 2 green. Why is the success probability about 0.005?
11. The server at deuce
In tennis the server wins each point with probability 0.6. Using the deuce formula w squared over w squared plus one minus w squared, compute the probability the server wins the game from a tied score and write it as a decimal.
12. Bob's share of the pennies
Bob plays matching pennies holding 15 of the 25 total pennies. Compute the probability he eventually wins all the pennies and write the answer as a fraction or decimal.
13. An uphill battle
A gambler wins one dollar with probability 0.4 and loses one with probability 0.6. Starting from 2 dollars, compute the probability of reaching 4 dollars before ruin using theta equal to q over p, and write the answer as a fraction or decimal.