All Questions
Tagged with probability-theory conditional-expectation
1,937
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Conditional expectation for a sum of iid random variables: $E(\xi\mid\xi+\eta)=E(\eta\mid\xi+\eta)=\frac{\xi+\eta}{2}$
I don't really know how to start proving this question.
Let $\xi$ and $\eta$ be independent, identically distributed random variables with $E(|\xi|)$ finite.
Show that
$E(\xi\mid\xi+\eta)=E(\eta\mid\...
- 171
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2
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Prove that $EX=E(E(X|Y))$
Prove that $EX=E(E(X|Y))$
I know that I should prove it from definition of conditional distribution and conditional expected value, but I don't know how.
I have also looked at theorem("Total ...
- 353
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2
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Proof of the tower property for conditional expectations
Let $Z$ be a $\mathfrak{F}$-measurable random variable with $\mathbb E(|Z|)<\infty$ and let $\mathfrak{H}\subset \mathfrak{G}\subset \mathfrak{F}$.
Show that then $\mathbb E(\mathbb E(Z|\mathfrak{...
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Conditional expectation with respect to a $\sigma$-algebra
Could someone explain what it is that we are intuitively trying to achieve with the definition? Having read the definition I could do the problems in the section of my book, but I still have no ...
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Conditional expectation as a Radon-Nikodym derivative.
I found the following very nice post yesterday which presented the conditional expectation in a way which I found intuitive;
Conditional expectation with respect to a $\sigma$-algebra.
I wonder if ...
user415535
10
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1
answer
4k
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Conditional expectation with the condition being a range
I can basically understand condition expectation with the condition being an event or a random variable($E(X|Y=y)$). However, I have a hard time understanding the condition being a range, especially ...
- 213
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1
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392
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What am I writing when I write $\mathbf X \mid \mathbf Y$?
Suppose $\mathbf X$ is a random variable and $A$ is an event in the same probability space $(\Omega, \mathcal F, \Pr)$. (Formally, $\mathbf X$ is a function on $\Omega$, say $\Omega \to \mathbb R$; $A$...
- 145k
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Rigorous definitions of probabilistic statements in Machine Learning
In a supervised machine learning setup, one usually considers an underlying measurable space $(\Omega, \mathcal{F}, \Bbb P)$ and random vectors/variables $X:\Omega \rightarrow \Bbb R^n, Y: \Omega \...
- 1,900
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1
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Exploiting the Markov property
I've encountered the following problem when dealing with short-rate models in finance and applying the Feynman-Kac theorem to relate conditional expectations to PDEs.
Let $(\Omega,\mathcal{F},\{\...
- 1,524
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802
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Conditional expectation continuous in the conditioning argument?
Let $X$ and $Y$ be random vectors defined on a common probability space. $X$ takes values in a finite-dimensional space $\mathcal{X} \subset \mathbb{R}^p$, while $Y$ takes values in $\mathbb{R}$. The ...
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Conditional expectation of random variable given a sum
Let $(X_i)_{i\geq1}$ i.i.d in $\mathcal{L}^1(\Omega,\mathcal{F},p)$ Is it true that
$E(X_j|\sum_{i=1}^nX_i)=\frac{1}{n}\sum_{i=1}^nX_i$
For each $j$ where $1\leq j \leq n$.
I think it is true, ...
- 874
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3
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768
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Expectation of X increases when conditioning on X being greater than another independent random variable Y?
I'm doing research and for a proof I need the smaller result that for X, Y random, independent (but not identical) variables we have
$$\mathbb{E}\left[X|X>Y\right] \geq \mathbb{E} \left[X\right]$$
...
- 91
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1
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Intuition for Conditional Expectation
It seems like NNT aka Nero in The Black Swan (2007) is giving the law of iterated expectations that involve filtrations in a heuristic way by matching the everyday usage of the word 'expect' with the ...
- 13.7k
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Existence of regular conditional distribution of random variable given the value of another variable
Let $(\Omega, \mathcal{A}, \mathbf{P})$ be a probability space with a measurable function $Y: (\Omega, \mathcal{A}) \rightarrow (E, \mathcal{E})$ and another measurable function $X: (\Omega, \mathcal{...
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If $Y\sim\mu$ with probability $p$ and $Y\sim\kappa(X,\;\cdot\;)$ otherwise, what's the conditional distribution of $Y$ given $X$?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(E,\mathcal E)$ be a measurale space
$\mu$ be a probability measure on $(E,\mathcal E)$
$X$ be an $(E,\mathcal E)$-valued random ...
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