Questions tagged [conditional-expectation]
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.
4,468 questions
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Inequality for central moments of bounded r.v. $\mu_3^2-2\mu_2^3\leq0$?
Let $(X_1,\dots,X_n,Y)$ be a random vector. Every $X_i$ takes values $-1,1$. Set
$$m(Y):=E[X_1|Y] = \int\dots\int x_1\;d P_{X|Y}(x_1,\dots,x_n|Y) $$
and suppose it satisfies $E[m(Y)^{2n}]=E[m(Y)^{2n-1}...
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About the conditional distribution
I am working in following setting: Let $\left(X_n\right)_{n \in \mathbb{N}}$ be a sequence of real, integrable, and independent random variables. Fix a $n \in \mathbb{N}$ and define $W_i := \sum_\...
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Conditional independence in Le Gall; is this claim true?
I'm working an exercise in Le Gall's Measure theory, probability and stochastic processes on conditional independence, Exercise 11.10. I'm paraphrasing some stuff:
Exercise 11.10. Let $\mathcal{B}$ ...
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Conditional convergence of uniformly integrable submartingale?
Let $X=(X_t)_{t\in\mathbb{N}}$ be a uniformly integrable submartingale
of a filtration $(\mathscr{F}_t)_{t\in\mathbb{N}}$.
Then the pointwise limit $X_\infty$ of $X$ exists and is finite almost surely....
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A concrete example of why we need regular conditional probability
There are perhaps some related questions on this site already (I can think of this question, this or this), but I want to be more concrete here. In the linked questions, and e.g. in Dudley's book (...
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Regular conditional probability being a measure almost everywhere; Le Gall
I have a great confusion with the definition of a regular conditional probability versus a section on the same subject I think in Le Gall's book Measure theory, probability and stochastic processes. ...
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Equality related to random variable and its conditional expectation.
Let $(\Omega,\mathcal F,\Bbb P)$ be a probability space and $X$ be a Banach space and $r_1,\cdots, r_N:\omega\to\Bbb R$ be random variables on the probability space, and $x_1,\cdots,x_N\in X.$
Suppose ...
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Expectation of an absolute value
Let $X,Y$ be two i.i.d.
I am trying to bound the expectation of how afar from one another they can get?
That is, $E[|X-Y|]$. I know that:
$$ E[X-Y] = E[X]- E[Y] = 0$$
But what about $|X-Y|$?
One ...
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Why does expectation project onto constant functions?
Let $X \in L^2$. Then $Z = E[X|G]$, for some sub $\sigma$-algebra $G$, is the orthogonal projection of $X$ onto $L^2(G)$. That is $Z$ is the random variable such that for every $G' \in G$:
$$\int_{G'} ...
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Prove a formula on conditional probability: $E(g(X,Y)| Y=y)= E(g(X,y)|Y=y)$.
Let $(\Omega,\mathcal F, P)$ be a probability space and $Z:\Omega\to\mathbb R$ be a random variable.
$Y: (\Omega,\mathcal F)\to (\Lambda,\mathcal G)$ is a measurable map between these two measurable ...
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Conditional expectation for non-integrable random variables
Let $(\Omega,\mathcal F,P)$ be a probability space, and let $\mathcal G\subset\mathcal F$ be a sub $\sigma$-algebra.
I am looking for a reference on defining $E[X|\mathcal G]$ with the most generality,...
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Expected payoff of the best lottery
Suppose that there are two lotteries: $1$ and $2$. The payoff of lottery $i \in \{1,2\}$, denoted by $u_i$, is a realization of an iid random draw from a compact interval $[0,1]$ according to a ...
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Tower Property implications on the conditional distribution
The tower property (law of total expectation) states that for any $σ$-subalgebras $G_1 ⊆ G_2$
$$ \text{(I)} \qquad E[X∣G_1] = E[E[X∣G_2] ∣ G_1] \qquad\text{a.s.} $$
In particular, for an integrable ...
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$L^2$ integrability of a regular version of a (gaussian"ish") conditional expectation
Preliminary note: I decided to post this question on MathOverflow too.
I am stuck on the following problem that in my opinion should definitely have a positive solution but every approach of mine has ...
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Follow-up to problem with my approach with "breaking stick at two points"
I'm trying to solve a "breaking a stick of length 1 at two points uniformly at random" problem. You are asked to find - with the same setup - the expected lengths of the shortest, middle, ...
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