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]
4197 questions
6
votes
1 answer
Covariance of conditional expectations
I'm interested in the relationship between $$cov(X,Y)$$ and $$cov\left(E(X|Y),E(Y|X)\right).$$ In particular, can it occur that $cov(X,Y)>0$ but $cov(E(X|Y),E(Y|X))<0$?
In the case of $\begin{bmatrix}X_1\\X_2\end{bmatrix}\sim…
stevo
- 61
5
votes
2 answers
Law of total probability in continous case
According to the law of total expectation, we have
$$\mathbb{E}X=\sum_{i=1}^n \mathbb{E}(X\mid A_{i})\cdot \mathbb{P}(A_i)$$
I am wondering if the similar formula holds for the continous case.
Specifically, if $Y$ is a continous random variable with…
Paweł Orliński
- 626
4
votes
1 answer
Increasing E(Y|X) implies increasing E(X|Y)
I guessed the following statement about Conditional Expectations and tried to prove it unsuccessfully:
if $E(Y|X=x)$ is strictly increasing in $x$, then $E(X|Y=y)$ is strictly increasing in $y$.
Any hint? I also tried to find a counter-example.
I…
AnonA
- 43
3
votes
1 answer
Conditional expectation: Prove $X_1 = X_2 \text{ a.s.}$
(This is a problem in chap 4 of 概率论教程（缪柏其 胡太忠著）. )
Set space $(\Omega, \mathscr{A}, P)$, $\mathscr{C}_1, \mathscr{C}_2 \subset \mathscr{A}$, and $\mathscr{C}_1, \mathscr{C}_2$ are both $\sigma$ algebra, $X\in L_1, X_1=E[X|\mathscr{C}_1],…
kashadong311
- 61
- 5
3
votes
1 answer
Conditional expectation for a weighted squared random variable
By the law of total expectation, the conditional expected value of X given Y is given by $$E(X)=E(E(X|Y))$$ Can someone please point me to the theory or identity showing the relation in the case we have the weighted squared shown below $$E(X^TQX)$$…
3
votes
0 answers
Why $\mathbb E\Big(\big(X-\mathbb E(X|Y )\big)^2\Big)\le 2\mathbb E\left(X^2\right)$?
Given $\mathbb E\left(X^2\right) \lt \infty $, why
$$\mathbb E\Big(\big(X -\mathbb E(X \mid Y )\big)^2\Big) \leq 2\mathbb E\left(X^2\right) \lt \infty \;\text?$$
Songlin Wei
- 31
3
votes
2 answers
Random variable related by conditional expectations
Let X and Y be random variables such that $E(X|Y)=\frac Y 2$ and $E(Y|X)=\frac X 2$. Does it follow that X and Y are 0? If not is their a simple example of such random variables?
Motivation: if $E(X|Y)= Y $ and $E(X|Y)=Y$ then X=Y necessarily. This…
Kavi Rama Murthy
- 311,013
3
votes
2 answers
is conditional expectation less than unconditional one?
Is that true that if $ F_{(k-1)}$ is a sigma algebra generated by random variables $X_1,X_2,\ldots,X_{k-1}$, then
$$ E(X_k \mid F_{k-1}) \le E(X_k)\quad\text{a.s. ?}$$
Anna
- 31
2
votes
0 answers
Is the equality $E(X|X^2)=X$ always true?
I know that $E(X^{2}|X)=X^2$ is true since $X^2$ is $\sigma(X)$-measurable. But I think that maybe $E(X|X^2)=X$ is not true since it could happen that $X$ be not $\sigma(X^2)$-measurable. But how to obtain a counterexample and, is my reasoning…
Iesus Dave Sanz
- 323
2
votes
0 answers
Exercise 5.29 of Introduction to Probability models by Ross.
This is exercise 5.29 of Introduction to Probability models by Ross.
Let X and Y be independent exponential random variables with respective rates $\lambda$ and $\mu$, where $\lambda > \mu$. Let $c>0$.
(a) Show that the conditional function of $X$,…
Ram Zi
- 67
2
votes
0 answers
Conditional expectation wrt a sub-$\sigma$-algebra
I am trying to wrap my head around conditional expectations wrt sub-$\sigma$-algebras. Sorry if this question is confusing, I tried my best!
I have trouble understanding the following illustration (from Wikipedia):
Here, the probability space is…
AlphaOmega
- 620
2
votes
1 answer
Partial expectation of a function of a lognormal random variable
Let $X$ be a lognormal random variable with pdf $f(X)$ and cdf $F(X)$.
The mean and variance of $X$ are assumed to be $\mu$ and $\sigma^2$ respectively.
If we assume any constant $k \ge 0$ and $0 < \alpha < 1$, how can we calculate following partial…
kang taesu
- 21
2
votes
2 answers
Find E(Y). Conditional Expectations
Let $X$ be an exponential random variable with $\lambda =5$ and $Y$ a uniformly distributed random variable on $(-3,X)$. Find $\mathbb E(Y)$.
My attempt:
$$\mathbb E(Y)= \mathbb E(\mathbb E(Y|X))$$
$$\mathbb E(Y|X) = \int^{x}_{-3} y \frac{1}{x+3}…
Sym
- 23
2
votes
2 answers
Proving that $E[XE[Y\mid G]] = E[YE[X\mid G]]$
Let $X$ and $Y$ be two bounded random variables in a probability space $(\Omega,\mathcal{F},P)$. $G$ is a sub $\sigma$-algebra on $\Omega$. I have to prove that:
$$E[XE[Y\mid G]] = E[YE[X\mid G]]$$
As I found in this question, it is quite easy to…
Broken_Window
- 546
2
votes
1 answer
Successive conditional expectations with respect to different sigma-algebras
I am new to the notion of conditional expectation w.r.t a sigma algebra.
If X is a random variable on probability space ($\Omega, \mathbb{F},\mathbb{P}$), where $\Omega$= {1,2,3} with $X(\omega)=\omega$. Take the uniform measure. Suppose…
Rann
- 719