For every question related to the concept of conditional expectation of a random variable with respect to a $\sigma$-algebra. It should be used with the tag (probability-theory) or (probability), and other ones if needed.

# Questions tagged [conditional-expectation]

3615 questions

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### Conditional expectation of a joint normal distribution

Let $X_1, X_2$ be jointly normal $N(\mu, \Sigma)$.
I know that in general, $\mathbb{E}[X_2|X_1]$ can be computed by integrating the conditional density, but in the case of jointly normal variables, it suffices to do a linear…

user357269

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### 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 $\mathcal{F} = \sigma H = \{\emptyset, H, H^c,…

Olorun

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### 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 continuous stochastic process $(X_t)$ together with…

Mots du Jour

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### 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 $\mathbb{E}[X|Y] = Y$ means
$$ \int_{\{Y\in{A}\}}X \, d\Bbb{P} =…

Latimer Leviosa

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### Conditional Expectation of X given X^2

What can we say about $E[X|X^2]$ in general? And if $X$ has density $f$ respect the Lebesgue measure?

user136725

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### Retrieve the random variable from its conditional expectations

I came across a problem that looks easy but turns out to be extremely hard. The problem goes as follows:
$X,Y$ are two independent random variables with support on interval $[0,1]$ and $\mathrm{E}[X]=\mathrm{E}[Y]=\mu \in (0,1)$. Construct a random…

Khánh Toàn

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### A winning wager that loses over time

This problem was posted in Scientific American (vol. 321.5, Nov 2019, p. 73), and it was troubling.
The game:
We flip a fair coin.
If we flip heads we gain 20% of our bet
If we flip tails we lose 17% of our bet.
Starting bankroll:…

Tom Boshoff

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### 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 $(\Omega, \mathcal{F}, \mathbb{P})$ refers to
probability without…

Clarinetist

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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 $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},\mathbb{P})$ be a filtered…

JohnSmith

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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 intuitive idea of what the definition is trying to…

JT1

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### 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 $Y, Y'$ both satisfy the criteria to be a conditional…

layman

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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{G})|\mathfrak{H})=\mathbb E(Z|\mathfrak{H})=\mathbb…

Epsilondelta

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### Conditional expectation to de maximum $E(X_1\mid X_{(n)})$

Let $X_1, \ldots, X_n$ a random sample of a Uniform(0,1):
Which is $E(X_1\mid X_{(n)})$ ?
where $X_{(n)}=\max\{X_1,\ldots,X_n\}$

rcg90

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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
$E(\xi\mid\xi+\eta)=E(\eta\mid\xi+\eta)=\frac{\xi+\eta}{2}$
Does anyone here have…

kkk

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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 $X:\Omega \rightarrow \Bbb R^n, Y: \Omega \rightarrow \Bbb R.$ We can then consider the…

John D

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