Proving Martingale Property and Stopping Theorem for Probability Homework

[dirk_mec1]

Homework Statement

http://img128.imageshack.us/img128/2010/95701129si7.png

The Attempt at a Solution

If I define $$Z_n = \frac{X_n}{n+2}$$ with X_n the number of white balls in stage n then how can I prove that it's martingale?

 

[e(ho0n3]

To start with, what is the definition of a martingale?

 

[dirk_mec1]

To start with, what is the definition of a martingale?

Do you know the answer to the question or are you teaching me?

$$E[ Z_{n+1}| Z_1 ,Z_2,...,Z_n] =Z_n$$

 

[pizzasky]

1) After the nth replication, there are $$Z_n(n+2)$$ white balls and $$(1-Z_n)(n+2)$$ black balls in the urn.

2) In the (n+1)th replication, there will either be a white ball or a black ball added into the urn. For each of these cases, what will the new proportion of white balls in the urn be and what are the associated probabilities? Use the definition of expectation to calculate $$E \ [\ Z_{n+1}\ | \ Z_{1} , \ Z_{2} \, ..., \ Z_{n} \ ]$$

 

[dirk_mec1]

1) After the nth replication, there are $$Z_n(n+2)$$ white balls and $$(1-Z_n)(n+2)$$ black balls in the urn.

2) In the (n+1)th replication, there will either be a white ball or a black ball added into the urn. For each of these cases, what will the new proportion of white balls in the urn be and what are the associated probabilities? Use the definition of expectation to calculate $$E \ [\ Z_{n+1}\ | \ Z_{1} , \ Z_{2} \, ..., \ Z_{n} \ ]$$

I thought that at each replication two white balls are added or removed. But suppose you're right then I get:

$$Z_{n+1}= \frac{Z_n(n+2)+1}{n+3}$$

$$Z_{n+1} = \frac{Z_n(n+2)-1}{n+3}$$

with probabilities 1/2 for each?

 

[pizzasky]

At each replication, a ball is drawn from the urn. The colour of the ball is noted, then that ball, together with a new ball of the same colour, are placed into the urn. Note then that after each replication, the number of white balls in the urn either stays the same or increases by one.

Now, what is the probability of drawing a white ball (similarly, a black ball) from the urn during the (n+1)th replication? (Not necessarily half!)

If a white ball is drawn during the (n+1)th replication, then that white ball will be put back into the urn and a new white ball added. So, what will the proportion of white balls be after the (n+1)th replication? Similarly, if a black ball is drawn during the (n+1)th replication, what will the new proportion of white balls in the urn be?

 

[dirk_mec1]

So

$$E[Z_{n+1}|Z_1,...,Z_n] = Z_n \cdot \frac{Z_n (n+2) +1}{n+3} +(1-Z_n) \cdot \frac{Z_n (n+2) +1}{n+3} =Z_n$$

Ok thanks. The next question is:

if T is the first stage at which a black ball is drawn prove that then:$$E \left[ \frac{1}{T+2} \right] - \frac{1}{4}$$ via the martingale stopping theorem.

http://img45.imageshack.us/img45/2362/85438730sk6.png If X_N is the number of white balls in stage n then:

$$E[Z_T] = E[Z_1] = \frac{1}{2} = E \left[ \frac{X_T}{T+2} \right] = E[X_T] E \left[ \frac{1}{T+2} \right]$$

Is this correct?