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
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A simple problem with conditional expectation
Let $x$ and $y$ two i.i.d having an uniform distribution over $[0,1]$. Then what is the conditional expectation, $\mathbb{E}[x / y\ |\ x < y]$.
It seems to me, this should be:
$\int_{\{x < y\}}{x/y\ dP(\cdot\ |\ x
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conditonal expectation second moment of a symmetric random variable
Given a random variable say $X$ which is symmetric about some point $a$.
If I define another random variable Y whose expectation $E[Y|X=x] = f(x)$.
Is there a mechanism to say something about the second moment $E[Y^2|X]$ in terms of X?
knk
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Positivity of conditional expectations
I am trying to prove the well-known positivity property of conditional expectations. Namely,
$X$ being a random variable on a measure space $(\Omega,\cal{F},P)$ and $\cal{A}$ being a sub-$\sigma$-algebra of $\cal{F}$, the following implication…
Calculon
- 5,725
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1 answer
$|X|$ is $\sigma(X^2)$-measurable
I am looking for a proof of the statement in the title. $X$ is a random variable defined on some measure space.
My thinking was as follows. This statement is equivalent to saying $\sigma(|X|) \subseteq \sigma(X^2)$. The definition is that…
Calculon
- 5,725
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How to calculate the following conditonal expectation?
I want to calculate the conditional person's correlation coefficient. But I don't know how to calculate the following expressions,especially the conditional expectation of $E[XY|X>=F^{-1}_X(p),Y>=F^{-1}_Y(q)]$. Who can help me? I want use their…
eric
- 83
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2 answers
Conditional expectation of standard normal distribution variables
The question is: Let $\mathrm {X} $, $\mathrm Y$ $\sim$ $\mathcal N(0,1)$. Knowing that they're independent variables find $\mathbb E(X^4+Y^3|X+Y)$
I did: $\mathbb E|X|=\mathbb E|Y|<\infty$, therefore $\mathbb E(X|X+Y)=\mathbb…
user201226
- 21
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1 answer
Prove $E(X|Y)=0$ given that $E[Xg(Y)]=0$ for any measurable function $g$.
Please help me prove $E(X|Y)=0$ given that, for any measurable function $g$:
$$E[Xg(Y)]=0$$
I have been trying using a definition of conditional expectation, but it does not seem to work.
Thanks!
bankrip
- 566
1
vote
1 answer
Compute conditional expectation
Let $f(x,y)=c(x^2+y^3)I_{[0,1]^2}(x,y)$ be the density of a random vector. Compute $\Bbb E [Z|X]$, where $Z=\frac{Y+1}{X+1}$.
My approach:
$f_x=\int_0^1…
Max
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conditional expectation related to an AR model
Suppose we have an AR model like:
$\dot{h}(t)=ah(t)+w(t)$
where $a<0$, $w(t)\sim \mathcal{CN}(0,1)$. My question is how to evaluate the following conditional expectation:
$\mathbb{E}\big[h(t) \big| \{h(s):0\leq s
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Conditional Expectation Query
I'm trying to read up on Conditional Expectation and have across the following equation, where X and Y are jointly discrete random variables:
$E[E[X|Y]] = E[X]$ which results in the Equation below. I don't understand how this equation was arrived at…
user131983
- 445
1
vote
1 answer
Conditional expectation on $\Omega = [0, 1]$ with lebesgue measure
Consider $\Omega=[0, 1]$ with $X(w) = \mathbb{1}_{[0, \frac{1}{3})}(w)+ 2\mathbb{1}_{[\frac{1}{3}, \frac{2}{3})}(w)$ and $Y(w) = 2w+1$. How do i calculate $E[X\mid Y]$ and $E[Y \mid X]$? I tried finding joint density of $(X, Y)$ and then using…
Kombajn
- 444
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vote
1 answer
Independent bidimensional random variables.
Let $(X_i,Y_i)$ be independent bidimensional random variables, is the following equality true?:
$$E\Big[X_1X_2\cdots X_n\mid Y_1,Y_2,\cdots,Y_n \Big]=\prod_1^n E\left[X_i\mid Y_i\right]$$
Speltzu
- 543
1
vote
1 answer
Convergence of conditional expectations $E[X|Y_i]$.
Let $X$ and $Y$ be random variables and suppose that the sequence $Y_i$ converges $L^1$ to $Y$:
$$Y_i\overset{L_1}{\longrightarrow}Y$$
Does $E[X|Y_i]$ also converge $L^1$ to $E[X|Y]$?:
$$E[X\mid Y_i]\overset{L_1}{\longrightarrow} E[X\mid Y]$$
Speltzu
- 543
1
vote
0 answers
Conditional expectation as the integral of cdf
If $X$ is a continuous random variable, how to show that
$$E[X|X>c] = \int_c^{\infty} (1-F_X(x))dx?$$
I showed
$$EX = E \int_0^ \infty 1_{X>x}dx = \int_0^\infty P(X>x)dx = \int_0^{\infty} (1-F_X(x))dx.$$
If it is given that $X>c$, does that mean…
Vika
- 447
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vote
1 answer
Understanding the Tower Property
I'm having trouble understanding the following I read from a book:
If $m < n$, then $E\left( E [ Y| F_n ] | F_m \right) = E [ Y| F_m ]$.
My arguments are as follows: say $Z_m = E [ Y| F_m ]$ and $Z_n = E [ Y| F_n ]$. $Z_m$ is $m$-measurable.…
mfnx
- 197