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

In summary, the conversation discusses the correct calculations for E(Sn-Sm) and E[(Sn-Sm)^2] as well as the incorrect calculation for E[(Sn-Sm)^3]. The correct calculation for E(Xm|Sn) is also discussed.
  • #1
WMDhamnekar
MHB
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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
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Are these above answers correct?
 
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  • #2
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.
 
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  • #3
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:
  • #4
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.
 
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  • #5
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?
 
  • #6
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.
 
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1. What is conditional expectation given ##\mathcal{F}_m##?

Conditional expectation given ##\mathcal{F}_m## is a statistical concept in which the expected value of a random variable is calculated based on a specific sub-sigma-algebra ##\mathcal{F}_m##. It is denoted as ##E(X|\mathcal{F}_m)## and represents the average value of the random variable X given the information contained in the sub-sigma-algebra ##\mathcal{F}_m##.

2. How is conditional expectation given ##\mathcal{F}_m## calculated?

The calculation of conditional expectation given ##\mathcal{F}_m## involves using the conditional probability formula: ##E(X|\mathcal{F}_m) = \frac{E(XI_A)}{P(A)}##, where A is an event in the sub-sigma-algebra ##\mathcal{F}_m## and ##I_A## is the indicator function for event A. This formula represents the expected value of X given that event A has occurred.

3. What is the significance of conditional expectation given ##\mathcal{F}_m##?

Conditional expectation given ##\mathcal{F}_m## is important in statistical analysis as it allows for the prediction of future outcomes based on known information. It is also used in various mathematical models and plays a key role in probability theory and stochastic processes.

4. How does conditional expectation given ##\mathcal{F}_m## differ from unconditional expectation?

The main difference between conditional expectation given ##\mathcal{F}_m## and unconditional expectation is that the former takes into account additional information from the sub-sigma-algebra ##\mathcal{F}_m##, while the latter does not. In other words, conditional expectation given ##\mathcal{F}_m## is a more specific and refined calculation of expected value compared to unconditional expectation.

5. Can conditional expectation given ##\mathcal{F}_m## be negative?

Yes, conditional expectation given ##\mathcal{F}_m## can be negative. This can occur when the random variable X has a negative value and the information contained in the sub-sigma-algebra ##\mathcal{F}_m## affects the expected value. It is important to note that the sign of conditional expectation given ##\mathcal{F}_m## does not necessarily reflect the sign of the random variable X itself.

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