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 question on conditional expectation of indicator variable
Suppose we have three random variables $Y$, $X$, and $Z$, each of which is univariate. Define an indicator function $1(|X-x|\leq a)=1$ if $x-a\leq X\leq x+a$ and $0$ otherwise, for some positive values $a$ and $x$.
Question: What is conditional…
Rico
- 11
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1 answer
Conditional expectations as functions
Consider the following conditional expectation:
$$
\mathbb{E}[f(X,Y,Z)\mid X,Y]
$$
I know that it can be written as $m(X,Y)$, where
$$
m(x,y)=\mathbb{E}[f(X,Y,Z)\mid X=x, Y=y].
$$
Is its section $y\mapsto m(X,y)$ equivalent to
$$
y\mapsto…
Robert W.
- 722
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1 answer
conditional expectation stationary processes
Lately I worked with stationary processes of the following kind. Let $(\varepsilon_i)_{i\in\mathbb{N}_0}$ be iid. RVs. Let $\xi_n := (...,\varepsilon_{n-1}, \varepsilon_n)$ and further let $g:\mathbb{R}^{\mathbb{N}_0} \to \mathbb{R}$ measurable such…
student7481
- 179
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1 answer
Problems when solving $E(X\mid Y)$
If we know $X\sim \operatorname{Pois}(\lambda)$, $Y\sim\operatorname{Pois}(\lambda_p)$:
When solving $E(X\mid Y)$, based on the law of iterated expectations, $E(X) = E(E(X\mid Y)) =\lambda$. And we know that $E(\lambda) = \lambda$, so can we just…
lemonnn29
- 11
1
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1 answer
Special case of the law of total expectation
Is this true : $E_{X|Y}[X|Y]=E_{Z|Y}[E_{X|Y,Z}[X|Y,Z]]$ ? I deduced it from the law of total expectation.
Thanks a lot !
corks__
- 21
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3 answers
Find $\mathbb E(X | X + 2Y)$, where $X$ and $Y$ ~ $U[0,1]$
Let $X$ and $Y$ two independent random variables with unifrom distribution $U[0,1]$. Need to find $\mathbb E(X | X + 2Y)$. Is my calculation below correct?
$\mathbb E(X | X + 2Y) = \mathbb E(X|X) + \mathbb E(X | 2Y) = X + 2\mathbb E(X|Y) = X +…
great_again
- 35
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0 answers
Conditional expectation given polynomials
Assume that $X\sim \mathcal{N}(0,\sigma_X^2)$ and $\epsilon \sim \mathcal{N}(0,\sigma_\epsilon^2)$ are independent and $k\in \mathbb{N}$.
Define $Y:= \sum_{i=1}^k \beta_i X^i + \epsilon$, where $\beta_1,...,\beta_k$ are real numbers.
Is it…
John
- 1,775
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- 27
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How to prove this propensity score weighting leads to averaged treatment effect among the treated?
Let $Y(1)$ and $Y(0)$ be the potential outcomes under the treatment and the control. $T$ represents the treatment status. We let
$ATT=E(Y(1)-Y(0)|T=1)$
Next, we let $e(x)$ denote the propensity score. One type of propensity score weighting scheme…
Jason
- 13
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1 answer
Conditional Expectation Formula on discrete random Variable Using Indicator Function
Conditioned on a discrete random variable, the conditional expectation is given by the formula :
$$E(X|Y=y)=\sum xp(x|Y=y)$$
However I've found another formula in Wikipedia that given an event H:
$$E(X|H)=\frac{E(X 1_H)}{p(H)}$$
Can anyone provide…
W.314
- 348
1
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0 answers
Conditional expectation of function of two independent random variables
Let $X,Y$ two real independent random variables. I need to compute
$$\mathbb E[e^\frac{X}{Y}|Y]$$
which as we know is a measurable function of $Y$. Now, for every $y$ fixed I am able to compute
$$\mathbb E[e^\frac{X}{y}]=\phi(y)$$
which is a…
Davide Maran
- 1,149
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1 answer
Where is the mistake in my idea? (Conditional Expectation)
Let $X_t$ be a stochastic process. $\mathcal{F}_t:=\sigma(X_s:0\leq s\leq t)$.
Suppose $\mathrm{E}[e^{ik(X_t-X_s)}|X_s]=e^{-\frac{1}{2}k^2(t-s)}$.
My idea
By the tower property of conditional expectation: if $\mathcal{H}$ is a sub…
sate
- 195
1
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0 answers
Joint expectation vs conditional expectations
Let $X$ and $Y$ two independent random variables with distribution $F_X$ and $F_Y$. Consider a function $g(X,Y) \in R$. Let $E_P\{\cdot\}$ denote expectation under the distribution $P$.
Am I correct that
\begin{equation}
E\{g(X,Y)\} = E_{F_Y}\{…
PinkCollins
- 21
1
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1 answer
$E(X^2|X-Y) E(X^3|X-2Y)$ for Gaussians?
For independent gaussians with following the normal distribution with expectation zero and variance one, how do I compute:
$E(X^2|X-2Y), E(X^3|X-2Y)$
I know that $X-2Y$,$X+2Y$ are independent. However, this does not seem to be enough to deduce the…
Dole
- 2,653
1
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1 answer
Dispersion of X about its conditional mean decreases as the σalgebra grows
I am just starting to learn some probability theory, so I apologize in advance if this is a trivial question.
Suppose $E[X^2] < \infty$ and define $Var(X|G) = E[(X − E[X|G])^2
|G]$.
Prove that the dispersion of $X$ about its conditional mean…
user548645
- 93
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0 answers
Conditional expectation of a function of two random variables given one of them
Given two random variables $X_1,X_2$ how does one prove $$E[g_1(X_1)g_2(X_2)|X_2] = E[g_1(X_1)|X_2]g_2(X_2) $$
I can see the intuition that since $X_2$ is given, the piece depending on it should come out of the expectation but I'm not being able to…
sixtyTonneAngel
- 1,098