Conditional expectation given ##\mathcal{F}_m##

[WMDhamnekar]

Homework Statement:

Suppose $X_1, X_2, \dots$ are independent random variables with $\mathbb{P}[X_j =1] = 1- \mathbb{P}[X_j =-1] =\displaystyle\frac13$ Let $S_n = X_1 +\dots + X_n$ and let
$\mathcal{F}_n$ denote the information contained in $X_1 , \dots , X_n$
1.If m < n Find $E[S_n |\mathcal{F}_m], E[S^2_n| \mathcal{F}_m], E[S^3_n |\mathcal{F}_m]$

2. If m < n Find $E [X_m| S_n ]$

Relevant Equations:

Not applicable

Are these above answers correct?

 

[mjc123]

You're getting careless again. In the first line you correctly say E(S_n-S_m) = (-1/3)(n-m).
So when later on you say S_mE(S_n-S_m) = -S_m/3, that's wrong, isn't it?
And E[(S_n-S_m)²] is not E(X_j²)(n-m). We worked this out in another thread, didn't we? Similarly with the cube.

 

[WMDhamnekar]

You're getting careless again. In the first line you correctly say E(S_n-S_m) = (-1/3)(n-m).
So when later on you say S_mE(S_n-S_m) = -S_m/3, that's wrong, isn't it?
And E[(S_n-S_m)²] is not E(X_j²)(n-m). We worked this out in another thread, didn't we? Similarly with the cube.

E(S_n - S_m)= -1/3 (n-m)
So, S_mE[S_n -S_m ] = -S_m(n-m)/3

Hence $E[S^2_n|\mathcal{F}_m] = S^2_m -\frac23 S_m (n-m) + (n-m) + \frac{(n-m)(n-m-1)}{9}$

$E[S^3_n|\mathcal{F}_m] = S^3_m - S^2_m(n-m) + S_m(n-m)\frac{(n-m)(n-m-1)}{3} -\frac{n-m}{3} - (n-m)(n-m-1)- \frac{(n-m)(n-m-1)(n-m-2)}{27}$
Are these above answer correct?
Note:

Now, do you mean to say E[(S_n - S_m)²] = (n- m) +$\frac{(n-m)(n-m-1)}{9}$
and E[(S_n - S_m)³] = $-\frac{n-m}{3} - (n-m)(n-m-1)- \frac{(n-m)(n-m-1)(n-m-2)}{27}$

 

[mjc123]

I think that's correct, except that the S_m term in the cube should be
3S_m{(n-m) + (n-m)(n-m-1)/9}
You multiplied when you should have added.

 

[WMDhamnekar]

I think that's correct, except that the S_m term in the cube should be
3S_m{(n-m) + (n-m)(n-m-1)/9}
You multiplied when you should have added.

Is my computed E(X_m| S_n] correct?

 

[mjc123]

You want E[X_m|S_n]. Why do you bring in S_m? You don't know that (in the terms of the question). I would say it's S_n/n.
S_n can take the values n-2k (0≤k≤n). If there are k values of -1 in the string, the probability of any one value being -1 is k/n, and the probability of it being 1 is (n-k)/n. The expectation of any X_i is then
1*(n-k)/n - 1*k/n = (n-2k)/n = S_n/n.