All Questions
Tagged with probability-theory conditional-expectation
1,931
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What allows me to write a nested expectation as a sum of expectations?
Assume I have a random process $x_1,x_2,\ldots,x_N$ defined on a probability space $(\Omega,G,P)$.
Assume I have the expectation
$$E_{x_1}\large[x_1+E_{x_2\vert x_1}\large[x_2+E_{x_3\vert x_1,x_2}\...
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Conditional expected value $ E[X|Y] $ is a function of $Y$
I would like to show or see a proof that the conditional expected value $ E[X|Y] $ is a function of $Y$ for arbitrary random variables $X,Y$ (especially not discrete or continuous).
I think for ...
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How to prove this formula about expectation ？
The continuous-time Markov chain has an infinitesimal generating element Q.
For all $f \gt$ $0$,define
$$Z(t)=f(X_t)\exp\left(-\int_0^t\left(\frac{Qf}{f}\right)(X_s)ds\right).$$
Define $\tau_n$ as the ...
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If $T$ is sufficient, then there's a probability $\mathbb{P}_0$ such [...] $\mathbb{E}_{\mathbb{P}_0}(X|T)=E_P(X|T)$ a.e. for all $P\in\mathfrak{M}$
Let $(\Omega,\Sigma)$ be a measurable space and $\mathfrak{M}$ a family of probability measures. Suppose that exists a $\sigma$-finite measure $\mu:\Sigma \to\overline{\mathbb{R}}$ such $P\ll \mu$ for ...
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Can we conclude that $X$ is $\mathcal{G}$-mensurable if $X=\mathbb{E}[X|\mathcal{G}]$ a.e.?
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, $\mathcal{G\subseteq F}$ be a sub-$\sigma$-algebra and $X:\Omega\to \mathbb{R}$ be an integrable random variable. Can we conclude that $X$ ...
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Alternative Proof of Hoeffding's Lemma / solve the equation $E[1_A | X] = (1+X)/2$ given $X$
In the context of proving the Hoeffding Lemma I came across a slightly weaker statement in the form of an exercise:
"If $X$ is a real valued random variable and $|X| \leq 1$ a.s. then there ...
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Definition of Left-Closable Martingale
I am currently studying martingales with Resnick's book A Probability Path. He defines a martingale as closed on the right if there is an $X \in L_1$ such that $X_n = \mathbb{E}[X \mid \mathcal{B}_n]$ ...
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conditional expectation of non-negative variable
In Angrist's book Mostly Harmless Econometrics, section 3.4.2, it says
$$ E[Y_i|D_i]=E[Y_i|Y_i>0,D_i]P[Y_i>0|D_i] $$, where $Y_i$ is a non-negative variable ($Y_i$ can be $0$) and $D_i$ is a ...
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Expectation of the squared conditional expectation
I am considering the expectation of the squared expectation, as asked here but with no answer and so wanted to get the communities thoughts.
Since $E[Y|X]$ is not independent with itself then the ...
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Conditional Expectation Notation in ARCH Model
I'm new to ARCH models, and I have a question about the correct notation for expressing the conditional expectation of the return at time $t(r_t)$ given the information available up to time t-1. I'd ...
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Iterated Conditional Expectation Problem Related to Random Walks with Retirement
I am currently working through the results presented in the paper titled "Consistent Price Systems and Face-lifting Pricing under Transaction Costs" and authored by P. Guasoni, M. Rásonyi ...
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Intuitively, why does the conditional expectation of X given the trivial sigma algebra equals the expected value of X itself?
If we are conditioning X given the trivial sigma algebra then we get the expectation of X, its proof is trivial but intuitively what does this case represent ?
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Conditional probability involving two random times, where only the distribution of one of them is used.
Consider two random times $\tau_{1}$ and $\tau_{2}$ defined on a common probability space $\left(\Omega, \mathcal{G}, \mathbb{Q}\right)$ with $\mathbb{Q}(\tau_{k}=0)=0$ and $\mathbb{Q}(\tau_{k}>t)&...
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What's the relationship between $(X_i)$ and $(X_i - \mathbb{E}[X_i | X_{<i}])$?
Let $X_1,\cdots,X_n$ be $n$ random variables on the same probability space $(\Omega, F, P)$, all with expectation $0$. Define $Y_i=X_i - \mathbb{E}[X_i | X_1, \cdots, X_{i-1}]$. Is it true that for ...
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Difficulty understanding the conditional expectation
I'm currently having a confusion dealing with the conditional expectation. Let's recall the definition first:
Let $X$ be an integrable function (or a random variable) defined on a probability space $(\...
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