All Questions
12
questions
17
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 ...
- 2,094
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
2
votes
0
answers
420
views
If $X$ is independent of $\mathcal{G}$ and $Y$ is $\mathcal{G}$-measurable, then $\mathbb{E}(\varphi(X, Y) | \mathcal{G}) = \mathbb{E}(\varphi(X, Y))$
I'm reading a proposition given without proof in this note.
Proposition 12.4. Let $\mathcal{G}$ be a sub-$\sigma$-field of $\mathcal{F}, X, Y$ be two random variables such that $X$ is independent of $...
- 5,817
11
votes
1
answer
1k
views
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
5
votes
0
answers
271
views
Sigma algebra generated by a homeomorphic random variable
Let $\Omega = [0,1]$ be our probability space with sigma algebra of borel sets on $[0,1]$ and Lebesgue measure on $[0,1]$. Let Y be a random variable such that $Y(\omega) = Y(1-\omega)$ for every $\...
- 2,601
5
votes
1
answer
973
views
Compute the conditional expectation $E(Y|X)$ for a measurable function $Y$ and a random variable $X$ taking values on $[0,1)$
Good day,
Currently I am working with "Probability: Theory and Examples" by Durrett and while getting familiar with conditional expectations I got to this problem:
Consider the Lebesgue ...
- 4,483
3
votes
1
answer
425
views
$X = E(Y | \sigma(X)) $ and $Y = E(X | \sigma(Y))$
Suppose $X, Y$ are random variables in $L^2$ such that $$X = E(Y | \sigma(X)) $$ $$Y = E(X | \sigma(Y))$$ Then I want to show that $X=Y$ almost everywhere.
What I've done:
By conditional Jensen $$...
- 11.5k
2
votes
1
answer
60
views
"co-relatedness" of conditional expectation of two independent random variables
Let $\mathcal{F}$ be a $\sigma$-algebra and $X,Y$ be two independent (not just uncorrelated) random variables, I wonder if the following statement true
$$\mathbb E(XY|\mathcal{F})=\mathbb E(X|\...
- 8,039
1
vote
1
answer
117
views
Elementary explanation of getting two consecutive $6$'s in a die roll experiment
I know that there are already numerous questions that adress this problem. However, I am not interested in a soltuion at all but in an explanation of a particular solution (see https://math....
- 4,564
1
vote
1
answer
2k
views
Conditional expectation in Poisson point process
Considering a Poisson process with parameter $\lambda$, let $N(t_2)$ denote the number of events in $(0,t_2]$ and $N(t_1, t_3)$ denote the number of events in $(t_1,t_3]$, under the assumption that $0&...
- 1,049
0
votes
1
answer
134
views
For $X,Y $ random variables, $h $ a function, show that $E (Xh(Y)|Y)=h (Y)E (X|Y) $ almost surely
Question in the title:
For $X,Y $ random variables, $h $ a function, show that $E (Xh(Y)|Y)=h (Y)E (X|Y) $ almost surely
My main problem is that I don't even understand what $E (Xh(Y)|Y)$ means... ...
- 7,305
0
votes
0
answers
661
views
A question on conditional expectation leading to zero covariance and vice versa
In my probability class I was tackled with this seemingly weird question involving conditional expectation:
Let X,Y be two random variables (it is not mentioned whether or not they are discrete or ...
- 3,847