All Questions
31
questions with no upvoted or accepted answers
5
votes
0
answers
271
views
Sigma algebra generated by a homeomorphic random variable
Let $\Omega = [0,1]$ be our probability space with sigma algebra of borel sets on $[0,1]$ and Lebesgue measure on $[0,1]$. Let Y be a random variable such that $Y(\omega) = Y(1-\omega)$ for every $\...
- 2,601
4
votes
0
answers
230
views
Calculation of Conditional Expectation $\Bbb E[f(X)\mid Y]$
$\newcommand{\Cov}{\operatorname{Cov}}$and thank you for taking the time to read this. :)
My question is about evaluating $\Bbb E[f(X) \mid Y]$ (a random variable in $Y$). There's plenty online (and ...
- 6,775
3
votes
0
answers
77
views
Correlation of conditional expectation of uncorrelated random variables
Let $X,Y\in\mathcal{L}_{2}\left(\Omega,\mathcal{F},\mathbb{P}\right)$ satisfy $\mathbb{E}\left[X\right]=\mathbb{E}\left[Y\right]=\mathbb{E}\left[XY\right]=0$, and $\mathbb{E}\left[X^2\right]=\mathbb{E}...
- 191
2
votes
0
answers
112
views
Conditional expectation of typos
I'm wondering how one could solve this problem:
A text consists of $n$ characters, each of which is a typo with probability $p$ (independently).
A proof reader then reads through the text and ...
2
votes
0
answers
420
views
If $X$ is independent of $\mathcal{G}$ and $Y$ is $\mathcal{G}$-measurable, then $\mathbb{E}(\varphi(X, Y) | \mathcal{G}) = \mathbb{E}(\varphi(X, Y))$
I'm reading a proposition given without proof in this note.
Proposition 12.4. Let $\mathcal{G}$ be a sub-$\sigma$-field of $\mathcal{F}, X, Y$ be two random variables such that $X$ is independent of $...
- 5,817
2
votes
0
answers
107
views
Computing a Conditional Expectation
Let $\pi$ be a permutation of ${1,\ldots ,n}.$ The pair $(i,j)$ with $i<j$ is an inversion if $\pi(i)>\pi(j).$ Denote the number of inversions by $\text{Inv}$. Let $M=(M_{i,j})$ be a $n\times n$ ...
- 3,918
2
votes
1
answer
110
views
Conditional expectation, specific function, three intervals
Let $\Omega= [0,1]$, $P$ be Lebesgue measure. Let $$Y(x) = \begin{cases} x^2, & \mbox{if } x \in [0, \frac{1}{3}) \\ \frac{1}{9}, & \mbox{if } x \in [\frac{1}{3}, \frac{2}{3}) \\ (x-1)^2, &...
- 363
2
votes
0
answers
32
views
Difference of random variables conditioned on their sum
Consider $\Omega = [0,1] \times [0,1]$ with sigma algebra of borel sets on $[0,1]^2$. Let $P$ be the Lebesgue measure on $\Omega$.
Let $$\xi(x, y) = x, \ \ \ \eta(x,y) = y.$$
How can I find $\mathbb{...
- 363
2
votes
0
answers
164
views
Density of a projection random variable
Let $\Omega = [0,1] \times [0,1]$ and let $\mathcal {B}([0,1] \times [0,1])$ be our sigma algebra.
Define two random variables: $\xi(x, y) = x \ $ and $ \ \eta (x,y) =y$.
Find $\mathbb{E}(\xi | \...
- 1,333
1
vote
0
answers
97
views
Find the conditional moments of a random variable
Suppose that we have three different variables $x$, $y$ and $z$, where $x$ stands for the state of the world and $\mathbb{X}$ is the state space, such that $x\in\mathbb{X}$. The following information ...
- 257
1
vote
0
answers
41
views
Independence of random variables meas. wrt independent sigma algebras.
Suppose we have two independent $\sigma$-algebras, $\mathcal{G}$ and $\mathcal{H}$. Let $X$ and $Y$ be two $(\mathcal{G}\cap\mathcal{H})$-measurable random variables. Then $\sigma(X)\subseteq\mathcal{...
- 1,023
1
vote
0
answers
39
views
How does conditional expectation tell the average of r.v. X on the union of some basic events?
Let $X\in L^1$ be a $\mathcal F$-measurable random variable and $\mathcal G$ be a sub σ-algebra of $\mathcal F$. We say a $\mathcal G$-measurable random variable $\mathbb E[X|\mathcal G]\in L^1$ is ...
- 57
1
vote
0
answers
383
views
Conditional Expectation Tower Property proof exercise
In the answer, i'm not sure how do derive the density of Y given Z. Also, why are we integrating with respect to y but not to z? I know by definition that if we're searching for expectation of Y given ...
user634512
1
vote
0
answers
417
views
Conditional expectation - conditioning with independent variables
I have doubts about two related equalities involving conditional expectation:
Let $X,Y,Z$ be random variables; let $Z$ be independent on $X,Y$.
Is $E[X \mid \beta Z + Y] = E[X \mid Y]$?
Is $E[\...
- 21.2k
1
vote
0
answers
199
views
Computing expectation of product of two random variables
Let $X$ be a random variable over $\mathbb{R}$ with density $P_X$. Assume a finite disjoint partition of $\mathbb{R}$, $Z_1,Z_2,\dots, Z_M$. That is,
\begin{equation}
\bigcup_i Z_i= \mathbb{R}\:\:\:\...
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