All Questions
Tagged with probability-theory conditional-expectation
1,937
questions
7
votes
2
answers
391
views
Expectation of maximum of arithmetic means of i.i.d. exponential random variables
Given the sequence $(X_n), n=1,2,... $, of iid exponential random variables with parameter $1$, define:
$$ M_n := \max \left\{ X_1, \frac{X_1+X_2}{2}, ...,\frac{X_1+\dots+X_n}{n} \right\} $$
I want ...
- 563
7
votes
1
answer
567
views
How to determine if conditional expectations with respect to different measures are equal a.s.?
Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $\mathcal{A}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. Let $Q_{\mathcal{A}}$ be a probability measure on $(\Omega, \mathcal{A})$ and ...
- 8,750
7
votes
1
answer
2k
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Inverse Mills ratio for non normal distributions.
We have the well known result of the inverse Mills ratio:
$$ \mathbb{E}[\,X\,|_{\ X > k} \,] =
\mu + \sigma \frac {\phi\big(\tfrac{k-\mu}{\sigma}\big)}{1-\Phi\big(\tfrac{k-\mu}{\sigma}\...
- 3,769
7
votes
1
answer
205
views
Conditional expectation $E[X_0|X_0X_1,\ldots, X_0X_n]$.
Consider iid random variables $(X_j)_{j\in\mathbb{N}_0}$ uniformly distributed on $[0,1]$.
For $j\in\mathbb{N}$ define $V_j:=X_0X_j$ and the recursively defined estimator $W_j:=\max (W_{j-1},V_j)$ ...
- 3,120
7
votes
1
answer
1k
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Hoeffding's inequality for conditional probability
I am currently reading the paper Functional Classification in Hilbert Spaces by Biau, Bunea and Wegkamp, and there is one step in the proof of Theorem 1 that is not clear to me. I give below a ...
7
votes
1
answer
665
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From disintegration to conditioning
There is a paper "Conditioning as disintegration" by J. T. Chang and D. Pollard, which seems to construct the regular conditional probability from the disintegration. In particular, from ...
- 1,972
7
votes
1
answer
726
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Is closeness in total variation preserved in conditioning?
If we have two joint distributions on $(X,Y)$, $P_{X,Y}$ and $Q_{X,Y}$ that are close in $L^1$ or "total variation" with $\|P_{X,Y}-Q_{X,Y}\|_1<\varepsilon$ then:
Are the distributions on the ...
- 4,886
7
votes
1
answer
4k
views
Derivative of conditional expectation
Let $\left( {{X_t}:t \in \left[ 0 \right.\left. {, + \infty } \right\rangle } \right)$ be a continuous time Markov chain on a probability space $\left( {\Omega ,\mathcal{F},\mathbb{P}} \right)$ with a ...
- 2,022
7
votes
1
answer
1k
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Is "conditional independence" of $\sigma$-algebras implied by "set-wise conditional independence" of $\sigma$-algebras?
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $\mathcal{G}_1,\mathcal{G}_2,\mathcal{H}$ be sub-$\sigma$-algebras of $\mathcal{F}$, and suppose that for all $G_1 \in \mathcal{G}_1$, ...
- 337
7
votes
1
answer
162
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If $f$ is a measurable random field, then $(ω,x)↦E[f(x)\mid F](ω)$ has a measurable version $g$ and $E[f(X)\mid F]=g(X)$ for all $F$-measurable $X$
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$\mathcal F\subseteq\mathcal A$ be a $\sigma$-algebra on $(\Omega,\mathcal A)$
$(E,\mathcal E)$ be a measurable space
$f:\Omega\times ...
- 13.9k
7
votes
3
answers
7k
views
conditional expectation given 2 random variables
Let's say there are random variables $A$ and $B$ being independent.
And random variable $X$.
Are there any properties to simplify $\mathbb{E}(X \mid (A,B))$ : expectation of $X$ given $A$ and $B$ ?
...
- 1,374
7
votes
1
answer
442
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Conditional Expectation with elliptical random variables
In a paper I am reading it is written the following:
Let $X = (X_1, \dots, X_n) \sim E_n(\mu, \Sigma, \phi)$ be a
elliptical-distributed random vector; let $S = \sum_{i = 1}^n X_i$.
Then $$E[...
- 21.2k
7
votes
1
answer
393
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Expectation of a quadratic form of Bernoulli random variables
Let $X_i$ be independent Bernoulli random variables with success probabilities $p_i$. That is, $\mathbb{P}(X_i = 1) = p_i, \mathbb{P}(X_i = 0) = 1-p_i$. Is there a closed expression or an approximate ...
- 85
7
votes
1
answer
317
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If $Y$ is a nonnegative absolutely continuous random variable and $E[X|Y]=Y/2$, is $E[X|Y=-1]=-1/2$? Is $E[X|Y=2]=1$?
One of the definitions I learned for $E[X|Y=y]$ is the following:
$$ E[X|Y=y]=\int_{\mathbb{R}} x\,P_{X|Y=y}(dx), $$
where $P_{X|Y=y}$ a probability verifying
$$ P(X\in A, Y\in B)=\int_B P_{X|Y=y}(A)\...
- 1,579
6
votes
2
answers
20k
views
What is the expectation of $X$ given $X$
Hi im trying to understand conditional expectation and conditional probability based on sigma algebras. Therefore an answer in that flavour would be most useful.
So in a physical sense I can see ...
- 2,320