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.
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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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Why does expectation project onto constant functions?
Let $X \in L^2$. Then $Z = E[X|G],ドル for some sub $\sigma$-algebra $G,ドル is the orthogonal projection of $X$ onto $L^2(G)$. That is $Z$ is the random variable such that for every $G' \in G$:
$$\int_{G'} ...
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Conditional expectation for non-integrable random variables
Let $(\Omega,\mathcal F,P)$ be a probability space, and let $\mathcal G\subset\mathcal F$ be a sub $\sigma$-algebra.
I am looking for a reference on defining $E[X|\mathcal G]$ with the most generality,...
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Tower Property implications on the conditional distribution
The tower property (law of total expectation) states that for any $σ$-subalgebras $G_1 ⊆ G_2$
$$ \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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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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Conditional independence for multiparameter filtration generated by independent filtrations
Let
$\{\mathscr{F}^1(n_1):n_1\in\mathbb{N}\}$
and
$\{\mathscr{F}^2(n_2):n_2\in\mathbb{N}\}$
be independent filtrations
in a probability space $(\Omega,\mathscr{F},\mathbb{P})$.
By defining for each $n ...
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An Exercise in Condition Distribution that does not Go Right
Consider the following exercise. Let $X$ be a random variable such that, knowing $\sigma > 0,ドル it has a conditional distribution of $N(0, \sigma^2)$. We give $\sigma$ a distribution of Lebesgue ...
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Is the conditional expectation of a partition really only defined up to $\mathbb{P}$ uniqueness?
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and let $(B_i)_i$ be a finite or countable partition of $\Omega,ドル that is, $\Omega = \cup_{i}B_i$ and $B_i\cap B_j=\emptyset$ for all $i\...