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

WMDhamnekar
MHB
Messages
376
Reaction score
28
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
1678086797503.png

1678086985664.png

1678087009376.png

1678087089479.png


Are these above answers correct?
 
Physics news on Phys.org
You're getting careless again. In the first line you correctly say E(Sn-Sm) = (-1/3)(n-m).
So when later on you say SmE(Sn-Sm) = -Sm/3, that's wrong, isn't it?
And E[(Sn-Sm)2] is not E(Xj2)(n-m). We worked this out in another thread, didn't we? Similarly with the cube.
 
  • Like
Likes WMDhamnekar
mjc123 said:
You're getting careless again. In the first line you correctly say E(Sn-Sm) = (-1/3)(n-m).
So when later on you say SmE(Sn-Sm) = -Sm/3, that's wrong, isn't it?
And E[(Sn-Sm)2] is not E(Xj2)(n-m). We worked this out in another thread, didn't we? Similarly with the cube.
E(Sn - Sm)= -1/3 (n-m)
So, SmE[Sn -Sm ] = -Sm(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[(Sn - Sm)2] = (n- m) +##\frac{(n-m)(n-m-1)}{9}##
and E[(Sn - Sm)3] = ##-\frac{n-m}{3} - (n-m)(n-m-1)- \frac{(n-m)(n-m-1)(n-m-2)}{27}##
 
Last edited:
I think that's correct, except that the Sm term in the cube should be
3Sm{(n-m) + (n-m)(n-m-1)/9}
You multiplied when you should have added.
 
  • Like
Likes WMDhamnekar
mjc123 said:
I think that's correct, except that the Sm term in the cube should be
3Sm{(n-m) + (n-m)(n-m-1)/9}
You multiplied when you should have added.
Is my computed E(Xm| Sn] correct?
 
You want E[Xm|Sn]. Why do you bring in Sm? You don't know that (in the terms of the question). I would say it's Sn/n.
Sn 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 Xi is then
1*(n-k)/n - 1*k/n = (n-2k)/n = Sn/n.
 
  • Like
Likes WMDhamnekar
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
Back
Top