y****n
发帖数: 60 | 1 fair coin
1) flip 100 times, if H you win 2$, if T you lose 1$, what is the E[gain]?
2) assume you start with 50$. you will stop either hit 0 or after 100 times
flip. calculate E[gain]
3) You can flip the coin 100 times, but instead of having a fixed stake, you
can freely choose the stake for each flip. Just before the flip, you start
with 100$. After each flip, if is comes up with H, you win twice your stake
(and your stake is returned). If it comes up with T, you lose your stake. i.
e. if you start with x and select a stake of s, then after the flip you will
either have x-s or x+2s. you can never make your stake larger than your
balance. Q: what is E[gain] and what is E[log(gain+100)]? | j*****4
发帖数: 292 | 2 gr的笔试题？
times
you
stake
i.
will
【在 y****n 的大作中提到】
: fair coin
: 1) flip 100 times, if H you win 2$, if T you lose 1$, what is the E[gain]?
: 2) assume you start with 50$. you will stop either hit 0 or after 100 times
: flip. calculate E[gain]
: 3) You can flip the coin 100 times, but instead of having a fixed stake, you
: can freely choose the stake for each flip. Just before the flip, you start
: with 100$. After each flip, if is comes up with H, you win twice your stake
: (and your stake is returned). If it comes up with T, you lose your stake. i.
: e. if you start with x and select a stake of s, then after the flip you will
: either have x-s or x+2s. you can never make your stake larger than your
| n*******t
发帖数: 67 | 3 第二题怎么做？用 martingale 可以看出 E(Gain) = E(T)/2，但是 E(T) 怎么求呢？ | g*********r
发帖数: 2847 | 4
E(gain)= E'(gain) * (1-P(fail to toss 100 times))-50*P(fail to toss 100 times)
where E'(gain) is the answer to question 1
【在 n*******t 的大作中提到】
: 第二题怎么做？用 martingale 可以看出 E(Gain) = E(T)/2，但是 E(T) 怎么求呢？
| e******o
发帖数: 757 | 5 第二题考古。 For a asymmetric random walk like this question, the
probability hitting -1 (starting from zero) is p=(sqrt(5)-1)/2, and p^X_t is
a martingale. Then P(fail)=[(sqrt(5)-1)/2]^50.You can construct a
martingale to calculate P(fail)
第三题考古没考到。只说best bet policy is kelly bet. | n****e
发帖数: 629 | 6 3): Strategy: Bet $100 at 1st time, if win, bet $300, ...
Reasoning: I'll bet all I have at last round. E(gain)=E($ before last round)
/2. Therefore I need to maximize E($ before last round), so I'll bet all I
have at second to last round...
times
you
stake
i.
will
【在 y****n 的大作中提到】
: fair coin
: 1) flip 100 times, if H you win 2$, if T you lose 1$, what is the E[gain]?
: 2) assume you start with 50$. you will stop either hit 0 or after 100 times
: flip. calculate E[gain]
: 3) You can flip the coin 100 times, but instead of having a fixed stake, you
: can freely choose the stake for each flip. Just before the flip, you start
: with 100$. After each flip, if is comes up with H, you win twice your stake
: (and your stake is returned). If it comes up with T, you lose your stake. i.
: e. if you start with x and select a stake of s, then after the flip you will
: either have x-s or x+2s. you can never make your stake larger than your
| n****e
发帖数: 629 | 7 3), 2nd q:Obviously after taking log(), the strategy is irrelevant to the
coin number. So find the x, 0<=x<=1, which maximize the objective function f
(x)=1/2ln(1+2x)+1/2ln(1-x). Then stick to this x on each round.
times
you
stake
i.
will
[发表自未名空间手机版 - m.mitbbs.com]
【在 y****n 的大作中提到】
: fair coin
: 1) flip 100 times, if H you win 2$, if T you lose 1$, what is the E[gain]?
: 2) assume you start with 50$. you will stop either hit 0 or after 100 times
: flip. calculate E[gain]
: 3) You can flip the coin 100 times, but instead of having a fixed stake, you
: can freely choose the stake for each flip. Just before the flip, you start
: with 100$. After each flip, if is comes up with H, you win twice your stake
: (and your stake is returned). If it comes up with T, you lose your stake. i.
: e. if you start with x and select a stake of s, then after the flip you will
: either have x-s or x+2s. you can never make your stake larger than your
| n****e
发帖数: 629 | 8 x=1/4. So you should bet 25% of the coins you have on each round.
f
[发表自未名空间手机版 - m.mitbbs.com]
【在 n****e 的大作中提到】
: 3), 2nd q:Obviously after taking log(), the strategy is irrelevant to the
: coin number. So find the x, 0<=x<=1, which maximize the objective function f
: (x)=1/2ln(1+2x)+1/2ln(1-x). Then stick to this x on each round.
:
: times
: you
: stake
: i.
: will
: [发表自未名空间手机版 - m.mitbbs.com]
| n*******t
发帖数: 67 | 9 p^X_t is a martingale 之前的我都理解，接下来为什么 P(fail)=[(sqrt(5)-1)/2]
^50？（这里 P(fail)是说在一百步之内就 hit 0 的概率么？）
is
【在 e******o 的大作中提到】
: 第二题考古。 For a asymmetric random walk like this question, the
: probability hitting -1 (starting from zero) is p=(sqrt(5)-1)/2, and p^X_t is
: a martingale. Then P(fail)=[(sqrt(5)-1)/2]^50.You can construct a
: martingale to calculate P(fail)
: 第三题考古没考到。只说best bet policy is kelly bet.
| e******o
发帖数: 757 | 10 就是hit 0的概率。不考虑100次的条件的话是这么多。
2]
【在 n*******t 的大作中提到】
: p^X_t is a martingale 之前的我都理解，接下来为什么 P(fail)=[(sqrt(5)-1)/2]
: ^50？（这里 P(fail)是说在一百步之内就 hit 0 的概率么？）
:
: is
| n*******t
发帖数: 67 | 11 这个对解决第二题没有用啊。那所以第二题到底该怎么做？
【在 e******o 的大作中提到】
: 就是hit 0的概率。不考虑100次的条件的话是这么多。
:
: 2]
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