s*******i
发帖数: 546 | 1 游戏规则：
最多抛100次硬币
如果正面，庄家给我一块钱
如果反面，我给庄家一块钱
庄家输到10块钱，游戏停止
How much should I pay for this game?
或者说庄家该为这个游戏定价多少？ | h**********e
发帖数: 44 | 2 0 if coin is fair.
If the coin is unfair, you need to know the boundary in your side and the
prob of head. | s*******i
发帖数: 546 | 3 Can you share your ideas in reaching this conclusion?
Thanks!
【在 h**********e 的大作中提到】
: 0 if coin is fair.
: If the coin is unfair, you need to know the boundary in your side and the
: prob of head.
| h**********e
发帖数: 44 | 4 If it is fair game, your asset is martingale (and so does that of your
counterpart). A martingale stopping at stopping time is also a martingale.
Your expectation is 0.
If it unfair, you need to pay (possibly negative) same amount for each toss,
but you need to know how many times (on average) are you going to play.
【在 s*******i 的大作中提到】
: Can you share your ideas in reaching this conclusion?
: Thanks!
| r********2
发帖数: 19 | 5 I agree it is a martingale, but the stopping time is not integrable.
My thought is,
P(tau<\infty)=1
therefore, to make the game fair you should pay 10 as premium.
【在 s*******i 的大作中提到】
: 游戏规则：
: 最多抛100次硬币
: 如果正面，庄家给我一块钱
: 如果反面，我给庄家一块钱
: 庄家输到10块钱，游戏停止
: How much should I pay for this game?
: 或者说庄家该为这个游戏定价多少？
| h**********y
发帖数: 41 | 6 我觉得interviewer是想考reflection principle。设t为抛硬币次数，M为player所得
，分两种情况：
（1） 先用reflection principle求100步内结束的概率P{t(M=10)<=100},这时
player所得必为10。
（2） 再求在100步时，player 所得为M的概率P{t(M=8) =100}，P{t(M=6) =100}，
P{t(M=4) =100}，P{t(M=2) =100}，P{t(M=-2) =100}，P{t(M=-4) =100}，…，P{t(M=
-100) =100}.
最后求总期望：
10×P{t(M=10)<=100}+[8C(8,100)+ 6C(6,100)+4(4,100)+2C(2,100)-2C(2,100)- 4(4,
100)-6(6,100)…-100(100,100)]/2^100; |
|
