All Questions
135
questions
0
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1
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61
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A question about the geometric interpretation of the conditional expectation
Let $X$ and $Y$ be two random variable. Suppose $E[X|Y]=0$, we have
$$E[XY]= E[\, E[ XY |Y ]\, ] = E[\, Y E[ X |Y ]\, ]= E[0] = 0$$
A geometric intuition also corroborates this fact if we think $E[X|...
- 351
5
votes
1
answer
132
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Sigma-algebra generated by conditional expectation
I am dealing with the following question: given two dependent random variables $X_1,X_2$, I am wondering whether the following equivalence for the generated sigma-algebras holds:
$$\sigma(X_1)=\sigma(...
- 95
1
vote
0
answers
97
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Find the conditional moments of a random variable
Suppose that we have three different variables $x$, $y$ and $z$, where $x$ stands for the state of the world and $\mathbb{X}$ is the state space, such that $x\in\mathbb{X}$. The following information ...
- 257
1
vote
1
answer
117
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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
2
votes
0
answers
112
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Conditional expectation of typos
I'm wondering how one could solve this problem:
A text consists of $n$ characters, each of which is a typo with probability $p$ (independently).
A proof reader then reads through the text and ...
2
votes
0
answers
420
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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
7
votes
4
answers
905
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Why is $E[X|X+Y] = E[Y |X+Y]$ if X,Y are i.i.d random variables
In proof of the fact that $E[X|X+Y] = \frac{X+Y}{2}$ when $X,Y$ are independent, identically distributed random variables, one uses the observation that $E[X|X+Y] = E[Y|X+Y]$ but I don't see why this ...
- 117
4
votes
1
answer
320
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Doob's Optional Stopping Theorem to find probabilities of stopping times
Suppose we have a simple random walk starting from $S_0=0$, and $S_n=X_1+\dots+X_n$ such that $$\mathbb{P}(X_i=1)=p \hspace{1em}\mathbb{P}(X_i=0)=r \hspace{1em} \mathbb{P}(X_i=-1)=q$$ for positive $p,...
- 115
0
votes
1
answer
157
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Substitute in what is known in conditional expectation
Motivation:
It might appear intuitive that $E(f(X,Y)|Y=y)=E(f(X,y)|Y=y)$, i.e. we just substitute in what is known in the conditional expectation. However, I want to prove this rigorously using the ...
- 167
0
votes
0
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279
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Definition of conditional expectation given an event
The conditional expectation of a random variable $X$ given a nonnegligible event $A$ is usually defined as $\mathbb{E}(X\mathbb{1}_A)/\mathbb{P}(A)$. How does one derive the fact that $\mathbb{E}(X|A)=...
- 1,022
0
votes
1
answer
210
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Property of conditional expectation $E(X | \mathcal{V})$ where $\mathcal{V}$ is $\sigma$-algebra.
I'm self-studying the probability theory, and I got stuck on the understanding of the definition given below and some consequences that follow from that definition.
Let $(\Omega, \mathcal{U}, P)$ be a ...
- 337
2
votes
1
answer
52
views
Showing expectation of a finite sum of a sequence of random variables, squared
I am working with Loeve's "On Almost Sure Convergence", specifically on the extension of Kolmogorov's inequality in Lemma 5.1.
As part of the proof, with the assumption $E(X_n|X_{n-1},...,...
- 35
3
votes
1
answer
62
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Equivalence of conditional expectations wrt two sigma-algebras
Let $X$ be a random variable and let $\mathcal{G}$, $\mathcal{H}$ be two sub-$\sigma$-algebras. Consider the equation $$\mathbb{E}(\mathbb{E}(X|\mathcal{G})|\mathcal{H}) = \mathbb{E}(X|\mathcal{G}\cap\...
- 1,023
1
vote
0
answers
41
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Independence of random variables meas. wrt independent sigma algebras.
Suppose we have two independent $\sigma$-algebras, $\mathcal{G}$ and $\mathcal{H}$. Let $X$ and $Y$ be two $(\mathcal{G}\cap\mathcal{H})$-measurable random variables. Then $\sigma(X)\subseteq\mathcal{...
- 1,023
3
votes
1
answer
41
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Find Conditional expectation of uniform variables ...
Let $\xi,\eta$ be independent random variables, both with uniform distribution on $[0,2]$. Find $E[\eta^2|\xi/\eta]$.
My attempt to solve the problem is in the attached file.
I believe I solved it, ...
