All Questions
Tagged with probability-theory conditional-expectation
1,937
questions
143
votes
6
answers
22k
views
Intuition behind Conditional Expectation
I'm struggling with the concept of conditional expectation. First of all, if you have a link to any explanation that goes beyond showing that it is a generalization of elementary intuitive concepts, ...
- 6,545
44
votes
8
answers
52k
views
Intuitive explanation of the tower property of conditional expectation
I understand how to define conditional expectation and how to prove that it exists.
Further, I think I understand what conditional expectation means intuitively. I can also prove the tower property, ...
- 14.7k
38
votes
4
answers
12k
views
If $E[X|Y]=Y$ almost surely and $E[Y|X]=X$ almost surely then $X=Y$ almost surely
Assume that $X$ and $Y$ are two random variables such that $Y=E[X|Y]$ almost surely and $X= E[Y|X]$ almost surely. Prove that $X=Y$ almost surely.
The hint I was given is to evaluate:
$$E[X-Y;X>a,...
- 1,975
24
votes
1
answer
627
views
Upper and Lower Bounds on $Var(Var(X\mid Y))$
Are there any particular properties that
\begin{align*}
Var(Var(X\mid Y))
\end{align*}
satisfies so that we can derive any upper and lower bounds on it.
For example, if we replace $Var$ with ...
- 6,015
22
votes
1
answer
6k
views
Fubini's theorem for conditional expectations
I need to prove that if $E \int_a^b |X_u|\,du = \int_a^b E|X_u|\,du$ is finite then:
$$E\left[\left.\int_a^b X_u\,du \;\right|\; \mathcal{G}\right] = \int_a^b E[X_u \mid \mathcal{G}]\,du.$$
I just ...
- 6,419
17
votes
2
answers
5k
views
Conditional expectation equals random variable almost sure
Let $X$ be in $\mathfrak{L}^1(\Omega,\mathfrak{F},P)$ and $\mathfrak{G}\subset \mathfrak{F}$.
Prove that if $X$ and $E(X|\mathfrak{G})$ have same distribution, then they are equal almost surely.
I ...
- 2,094
14
votes
4
answers
6k
views
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 $\...
- 1,561
14
votes
1
answer
12k
views
Independence and conditional expectation
So, it's pretty clear that for independent $X,Y\in L_1(P)$ (with $E(X|Y)=E(X|\sigma(Y))$), we have $E(X|Y)=E(X)$. It is also quite easy to construct an example (for instance, $X=Y=1$) which shows that ...
- 299
14
votes
1
answer
8k
views
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 ...
- 175
13
votes
2
answers
966
views
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{...
- 19.6k
12
votes
1
answer
830
views
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 ...
- 137
11
votes
2
answers
9k
views
What is $E(X\mid X>c)$ in terms of $P(X>c)$?
What is $E(X\mid X>c)$ in terms of $P(X>c)$?
I've seen conditional probability/expectation before with respect to another random variable but not to the variable itself. How would I go about ...
- 193
11
votes
1
answer
540
views
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$ ...
- 797
11
votes
1
answer
1k
views
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 ...
- 115
11
votes
1
answer
2k
views
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,...
- 20.4k
10
votes
4
answers
5k
views
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\...
- 171
10
votes
2
answers
51k
views
Prove that $EX=E(E(X|Y))$
Prove that $EX=E(E(X|Y))$
I know that I should prove it from definition of conditional distribution and conditional expected value, but I don't know how.
I have also looked at theorem("Total ...
- 353
10
votes
2
answers
29k
views
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{...
- 515
10
votes
2
answers
3k
views
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 ...
- 653
10
votes
2
answers
5k
views
Conditional expectation as a Radon-Nikodym derivative.
I found the following very nice post yesterday which presented the conditional expectation in a way which I found intuitive;
Conditional expectation with respect to a $\sigma$-algebra.
I wonder if ...
user415535
10
votes
1
answer
4k
views
Conditional expectation with the condition being a range
I can basically understand condition expectation with the condition being an event or a random variable($E(X|Y=y)$). However, I have a hard time understanding the condition being a range, especially ...
- 213
10
votes
1
answer
392
views
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$...
- 145k
10
votes
2
answers
360
views
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 \...
- 1,900
10
votes
1
answer
234
views
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},\{\...
- 1,524
10
votes
0
answers
802
views
Conditional expectation continuous in the conditioning argument?
Let $X$ and $Y$ be random vectors defined on a common probability space. $X$ takes values in a finite-dimensional space $\mathcal{X} \subset \mathbb{R}^p$, while $Y$ takes values in $\mathbb{R}$. The ...
9
votes
2
answers
8k
views
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, ...
- 874
9
votes
3
answers
768
views
Expectation of X increases when conditioning on X being greater than another independent random variable Y?
I'm doing research and for a proof I need the smaller result that for X, Y random, independent (but not identical) variables we have
$$\mathbb{E}\left[X|X>Y\right] \geq \mathbb{E} \left[X\right]$$
...
- 91
9
votes
1
answer
1k
views
Intuition for Conditional Expectation
It seems like NNT aka Nero in The Black Swan (2007) is giving the law of iterated expectations that involve filtrations in a heuristic way by matching the everyday usage of the word 'expect' with the ...
- 13.7k
9
votes
2
answers
2k
views
Existence of regular conditional distribution of random variable given the value of another variable
Let $(\Omega, \mathcal{A}, \mathbf{P})$ be a probability space with a measurable function $Y: (\Omega, \mathcal{A}) \rightarrow (E, \mathcal{E})$ and another measurable function $X: (\Omega, \mathcal{...
- 93
9
votes
2
answers
426
views
If $Y\sim\mu$ with probability $p$ and $Y\sim\kappa(X,\;\cdot\;)$ otherwise, what's the conditional distribution of $Y$ given $X$?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(E,\mathcal E)$ be a measurale space
$\mu$ be a probability measure on $(E,\mathcal E)$
$X$ be an $(E,\mathcal E)$-valued random ...
- 13.9k
8
votes
2
answers
491
views
Conceptual Issues in the Measure Theoretic Proof of Conditional Expectations (via Radon-Nikodym)
I have been looking into measure theory (from a probabilist's perspective), and I have found the proof of the existence of the conditional expectation to feel a little "glossed over" in ...
- 912
8
votes
1
answer
4k
views
Radon-Nikodym-derivative as a martingale
At the beginning of all the stuff about Girsanov theorem, we introduced the Radon-Nikodym derivative as $Z_\infty := \frac{d \mathbb{Q}}{d \mathbb{P}}\vert_{\mathcal{F}_\infty}$.
Next, we considered ...
- 728
8
votes
5
answers
2k
views
Conditional expectation of independent variables
Claim. Let $Z_1, Z_2$ be two independent and identically distributed random variables. Then we have:
$$
\mathbb E[Z_1|Z_1+Z_2] =\frac{Z_1+Z_2}{2}.
$$
Proof.
To see this, I have proceeded as follows. ...
- 1,407
8
votes
1
answer
2k
views
Conditional expectation of $X$ given $|X|$
Let $X$ be in integrable, with density $f$ with respect to the Lebesgue measure.
Compute the conditional expectation : $ \operatorname{E} \left[ X\, \Big|\, |X| \,\right] $
My ansatz was : $ \...
- 83
8
votes
1
answer
2k
views
Conditional expectation and independence on $\sigma$-algebras and events
In many statistics papers, proofs might proceed as follows: Under the event $A$, the random variables $X$ and $Y$ are independent. (Often this means that on $A^C$, they might be dependent). Then some ...
- 2,822
8
votes
2
answers
199
views
Find the conditional expectation $E(U\mid \min(U,1−U))$ where $U∼U[0,1]$
Assume that $U$ is a random variable on $(Ω,\mathcal{F},P)$ with $U∼U[0,1]$. Let $Y= \min(U,1−U)$. I am asked to find $E(U\mid Y)$.
I have figured out a proof for this and I want some sanity check.
My ...
- 587
8
votes
1
answer
7k
views
Polya's urn (martingale)
Suppose you have an urn containing one red ball and one green ball. You draw one at random; if the ball is red, put it back in the urn with an additional red ball , otherwise put it back and add a ...
- 273
8
votes
2
answers
376
views
Show that $\mathbb{E}\left(\bar{X}_{n}\mid X_{(1)},X_{(n)}\right) = \frac{X_{(1)}+X_{(n)}}{2}$
Let $X_{1},\ldots,X_{n}$ be i.i.d. $U[\alpha,\beta]$ r.v.s., and let $X_{(1)}$ denote the $\min$, and $X_{(n)}$ the $\max$. Show that
$$
\mathbb{E}\left(\overline{X}_{n}\mid X_{(1)},X_{(n)}\right) = ...
- 433
8
votes
1
answer
290
views
Show that $E(X\mid Y, Z) = E(X\mid Y)$ almost surely with condition Z is independent of $(X, Y)$
$(X, Y, Z)$ is a continuous random vector and $Z$ is independent of $(X,Y)$. Prove that $E(X\mid Y, Z) = E(X\mid Y)$ almost surely.
I had been thinking this question tonight but couldn't figure out ...
- 198
8
votes
3
answers
580
views
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 ...
- 2,187
8
votes
0
answers
242
views
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 ...
- 115
7
votes
2
answers
3k
views
Holder conditional inequality
we consider, on a probability space $(\Omega,\mathcal{A},P)$, two random variable $X$ and $Y$ and let $\mathcal{H} \subset \mathcal{A}$ be a $\sigma$-algebra. Let $p,q>1$ such that $\frac{1}{p}+\...
user666433
7
votes
1
answer
651
views
Rigorous definition of the conditional expectations $E(X|Y=y)$ when $P(Y=y)=0$
Let $X$ be an integrable random variable on $(\Omega, \mathfrak A, P)$.
I've learned that for an event $A$ of non-zero probability,
$$
E(X|A) = \int X(\omega) \,dP(\omega|A) = \frac{1}{P(A)}\int_A X ...
- 3,246
7
votes
1
answer
2k
views
Joint densities and conditional densities of sums of i.i.d. normally distributed random variables
Let $X_1,X_2,…$ be independent with the common normal density $\eta$, and $S_k= X_1+⋯+X_k$. If $m <n$ find the joint density of $(S_m,S_n)$ and the conditional density for $S_m$ given that $S_n=t$.
...
- 5,768
7
votes
4
answers
905
views
Why is $E[X|X+Y] = E[Y |X+Y]$ if X,Y are i.i.d random variables
In proof of the fact that $E[X|X+Y] = \frac{X+Y}{2}$ when $X,Y$ are independent, identically distributed random variables, one uses the observation that $E[X|X+Y] = E[Y|X+Y]$ but I don't see why this ...
- 117
7
votes
2
answers
391
views
Expectation of maximum of arithmetic means of i.i.d. exponential random variables
Given the sequence $(X_n), n=1,2,... $, of iid exponential random variables with parameter $1$, define:
$$ M_n := \max \left\{ X_1, \frac{X_1+X_2}{2}, ...,\frac{X_1+\dots+X_n}{n} \right\} $$
I want ...
- 563
7
votes
1
answer
567
views
How to determine if conditional expectations with respect to different measures are equal a.s.?
Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $\mathcal{A}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. Let $Q_{\mathcal{A}}$ be a probability measure on $(\Omega, \mathcal{A})$ and ...
- 8,750
7
votes
1
answer
2k
views
Inverse Mills ratio for non normal distributions.
We have the well known result of the inverse Mills ratio:
$$ \mathbb{E}[\,X\,|_{\ X > k} \,] =
\mu + \sigma \frac {\phi\big(\tfrac{k-\mu}{\sigma}\big)}{1-\Phi\big(\tfrac{k-\mu}{\sigma}\...
- 3,769
7
votes
1
answer
205
views
Conditional expectation $E[X_0|X_0X_1,\ldots, X_0X_n]$.
Consider iid random variables $(X_j)_{j\in\mathbb{N}_0}$ uniformly distributed on $[0,1]$.
For $j\in\mathbb{N}$ define $V_j:=X_0X_j$ and the recursively defined estimator $W_j:=\max (W_{j-1},V_j)$ ...
- 3,120
7
votes
1
answer
1k
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Hoeffding's inequality for conditional probability
I am currently reading the paper Functional Classification in Hilbert Spaces by Biau, Bunea and Wegkamp, and there is one step in the proof of Theorem 1 that is not clear to me. I give below a ...
