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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### Simple random walk on cycle graph (Ending on specific vertex after cover time)

I'm considering a simple random walk on a cycle graph comprising a number of vertices, labelled $1$ to $5$ consecutively. Suppose I start at vertex 1 and can traverse to either side ($2$ or $5$). I continue this random walk until I have covered all…

Ice Tea

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### How can two seemingly identical conditional expectations have different values?

Background
Suppose that we are using a simplified spherical model of the Earth's surface with latitude $u \in (-\frac {\pi} 2, \frac {\pi} 2)$ and longitude $v \in (-\pi, \pi)$. Restricting attention to the hemisphere, $H$, where $u, v \in (-\frac…

Ethan Mark

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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 $\Omega \to \mathbb R$; $A$ is a subset of $\Omega$.)
I am comfortable…

Misha Lavrov

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### 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 suppose that $ X_{n } \to X $ a.s. and in $…

Pass Stoneke

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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 variable on $(\Omega,\mathcal A,\operatorname…

0xbadf00d

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### Finding expected value of remaining piece

A father has a pie made for his two sons. Eating more than half of the pie will give indigestion to anyone. While he is away, the older son helps himself to a piece of the pie. The younger son then comes and has a piece of what is left by the…

Mr. Bromwich I

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### Some version of Itô isometry with conditional expectations

Let $B = (B_t)_{t \geq 0}$ be a Brownian motion, $ \mathcal{F}=
(\mathcal{F}_t)_{t \geq 0}$ the natural filtration associated to $B$, $u \in L^2_{a,T}$ (that is, $u$ is an stochastic process $u = (u_t)_{0 \leq t \leq T}$ adapted to $\mathcal{F}$ so…

Alejandro Molero

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### Expectation of quotient of random variables

Let $X_1,...X_n$ be independent, identically distributed and nonnegative random variables, and let $k\le n$. Compute: $$E\left[{\sum_{i=1}^k X_i\over \sum_{i=1}^n X_i}\right].$$ This question has already been asked: Expectation of random variables…

user128422

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### The existence of conditional expectation with respect to a sub-$\sigma$-algebra

I was trying to solve the exercise 3.17 from the book of real analysis by Folland and I've found a problem. The first part of the exercise is the following:
Let $(X, M, \mu) $ be a $\sigma$-finite measure space, $ N $ a sub-$\sigma $-algebra of M…

mathnewbie

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### $P(X_{(n-k_n)}>X_1\mid X_1>u_n)=0$?

Let $X_1,X_2,\dots$ be continuous random variables with full support (I need the result when they follow AR time series $X_i=\alpha X_{i-1}+\varepsilon_i$ for iid epsilons. But if you will consider iid case, it may also help with the time series…

Albert Paradek

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### Regular conditional probability on Polish space and absolute continuity

Let $(\Omega,\mathcal F,\mathbb P)$ is a standard Borel space (i.e. $\Omega$ is Polish and $\mathcal F = \mathcal B(\Omega)$).
Then $\mathcal F$ is separable and for every sub-sigma-algebra $\mathcal G \subset \mathcal F$, then there exists a…

Cyril B.

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### Probability and expected steps for two ants to meet on cube

Here I present an extension to the famous ant on a cube question:
Two ants, A and B, are placed on diametrically opposite corners of a
cube. With every step, each ants move from one vertex to an adjacent vertex (with 1/3 probability of moving…

user107224

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### Multi-Gaussian Integrals with Heaviside for cosmic connectivity

Context
I would like to
predict the connectivity of the so-called cosmic web in arbitrary dimensions.
This is the cosmic web (in a hydrodynamical simulation)
The little wiggly things are galaxies (as traced by the gas density) while the inset…

chris

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### Conditional expectation of product of conditionally independent random variables

I would like to show the following statement using the general definition of conditional expectation. I believe it is true as it was also pointed out in other posts.
Let $X,Y$ be conditionally independent random variables w.r.t a sigma algebra…

User0000

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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, because given the information of the sum the best…

Daniel Ordoñez

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