All Questions
Tagged with probability-theory conditional-expectation
1,931
questions
142
votes
6
answers
22k
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Intuition behind Conditional Expectation
I'm struggling with the concept of conditional expectation. First of all, if you have a link to any explanation that goes beyond showing that it is a generalization of elementary intuitive concepts, ...
- 6,505
44
votes
8
answers
51k
views
Intuitive explanation of the tower property of conditional expectation
I understand how to define conditional expectation and how to prove that it exists.
Further, I think I understand what conditional expectation means intuitively. I can also prove the tower property, ...
- 14.7k
38
votes
4
answers
12k
views
If $E[X|Y]=Y$ almost surely and $E[Y|X]=X$ almost surely then $X=Y$ almost surely
Assume that $X$ and $Y$ are two random variables such that $Y=E[X|Y]$ almost surely and $X= E[Y|X]$ almost surely. Prove that $X=Y$ almost surely.
The hint I was given is to evaluate:
$$E[X-Y;X>a,...
- 1,975
24
votes
1
answer
626
views
Upper and Lower Bounds on $Var(Var(X\mid Y))$
Are there any particular properties that
\begin{align*}
Var(Var(X\mid Y))
\end{align*}
satisfies so that we can derive any upper and lower bounds on it.
For example, if we replace $Var$ with ...
- 6,015
22
votes
1
answer
6k
views
Fubini's theorem for conditional expectations
I need to prove that if $E \int_a^b |X_u|\,du = \int_a^b E|X_u|\,du$ is finite then:
$$E\left[\left.\int_a^b X_u\,du \;\right|\; \mathcal{G}\right] = \int_a^b E[X_u \mid \mathcal{G}]\,du.$$
I just ...
- 6,399
16
votes
2
answers
5k
views
Conditional expectation equals random variable almost sure
Let $X$ be in $\mathfrak{L}^1(\Omega,\mathfrak{F},P)$ and $\mathfrak{G}\subset \mathfrak{F}$.
Prove that if $X$ and $E(X|\mathfrak{G})$ have same distribution, then they are equal almost surely.
I ...
- 2,084
14
votes
4
answers
6k
views
Conditional expectation given an event is equivalent to conditional expectation given the sigma algebra generated by the event
This problem is motivated by my self study of Cinlar's "Probability and Stochastics", it is Exercise 1.26 in chapter 4 (on conditioning).
The exercise goes as follows: Let H be an event and let $\...
- 1,561
14
votes
1
answer
12k
views
Independence and conditional expectation
So, it's pretty clear that for independent $X,Y\in L_1(P)$ (with $E(X|Y)=E(X|\sigma(Y))$), we have $E(X|Y)=E(X)$. It is also quite easy to construct an example (for instance, $X=Y=1$) which shows that ...
- 299
14
votes
1
answer
8k
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Conditional expectation of product of conditionally independent random variables
I would like to show the following statement using the general definition of conditional expectation. I believe it is true as it was also pointed out in other posts.
Let $X,Y$ be conditionally ...
- 175
13
votes
2
answers
960
views
What is the intuition behind conditional expectation in a measure-theoretic treatment of probability?
What is the intuition behind conditional expectation in a measure-theoretic sense, as opposed to a non-measure-theoretic treatment?
You may assume I know:
what a probability space $(\Omega, \mathcal{...
- 19.6k
12
votes
1
answer
830
views
Conditional expectation $\mathbb E\left(\exp\left(\int_0^tX_sdB_s\right) \mid \mathcal F_t^X\right)$
I have found a theorem (see below) in two papers an I try to figure how it could be proved. The result seems to be intuitive, but I'm not able to prove it in a rigorous way.
Assumptions:
Consider a ...
- 137
11
votes
2
answers
9k
views
What is $E(X\mid X>c)$ in terms of $P(X>c)$?
What is $E(X\mid X>c)$ in terms of $P(X>c)$?
I've seen conditional probability/expectation before with respect to another random variable but not to the variable itself. How would I go about ...
- 193
11
votes
1
answer
536
views
Given $\mathbb{E}[X|Y] = Y$ a.s. and $\mathbb{E}[Y|X] = X$ a.s. show $X = Y$ a.s.
Given $X,Y \in L^2(\Omega,\mathscr{F},\Bbb{P})$ such that
$\mathbb{E}[X|Y] = Y$ a.s.
$\mathbb{E}[Y|X] = X$ a.s.
show that $\Bbb{P}(X = Y ) = 1.$
$Attempt: $
I can see that $\mathbb{E}[X|Y] = Y$ ...
- 797
11
votes
1
answer
1k
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Does almost sure convergence and $L^1$-convergence imply almost sure convergence of the conditional expectation?
Question. Let $ X_{n}, X $ be random variables on some probability space $ ( \Omega, \mathcal{F},\mathbb{P} ) $ and let $ \mathcal{G} \subset \mathcal{F} $ be a sub-$\sigma$-algebra. Moreover ...
- 115
11
votes
1
answer
2k
views
Proof of uniqueness of conditional expectation
I have a question on the proof Durrett (p. $190$) gives for the uniqueness of the conditional expectation function.
If I understand his proof correctly, here is what I think it is saying:
Suppose $Y,...
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