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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Conditional Expectation (orthogonal projection): Why unique minimizer?
Let $(\Omega, \mathcal{A}, P)$ be a probability space and consider a square-integrable random variable $X$ and a sub-sigma-Algebra $\mathcal{F}$.
Denote $Z:=E[X \mid \mathcal{F}]$.
Conditional expectation is the unique minimizer (orthogonal…
user146358
- 309
1
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0 answers
Conditional expectation on two variables to conditional expectation in 1 variable
I am really confused trying to prove the following statement formally although it makes perfect intuitive sense.
I want to show that if $Y$ is conditionally independent on $W$ given $X$, then:
$E[W|X]=E[W|X,Y]$
I know I have to use the Law of…
1
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1 answer
Find unconditional distribution (using conditional expectation)
$f(Θ)$ is pdf of gamma distribution
$$f(Θ) = \frac{λ^α}{Γ(α)}Θ^{α-1}\exp(-λΘ), $$
$$X\mid Θ \sim \mathrm{poisson}(Θ) \rightarrow \frac{Θ^x\exp(-Θ)}{x!}$$
Suppose that $Θ$ is a random variable that follows a gamma distribution with parameters $λ$…
Sung Pang
- 27
1
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2 answers
A problem on conditioning where I am completely lost.
The real random variables $X$ and $Y$ are independent and both have a uniform distribution $U([0,1])$. Find
$$\mathbb{E}\left[e^{X+Y}|\quad |X-Y| \right]$$
Since $f(\cdot)=|\cdot|$ is not a monotone function, we can't remove it. I.e. if we had…
Vadim Omelchenko
- 219
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1 answer
How to show $E(XE(Y\mid F)=E(E(X\mid F)Y)$?
Let $X,Y\in\mathcal{L}^2$ and let $F$ be a $\sigma$-algebra.
How to show that $E(XE(Y\mid F)=E(E(X\mid F)Y)$?
Maybe you can give me some help?
Rhjg
- 2,029
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1 answer
Is second moment of exponential also Memory Less?
I know that the exponential is memory less and that means that:
$$
E[X\mid X>1]=1+E[X]
$$
Now, does the memory less property also hold for the second moment?
Specifically, is the following true? $$
E[X^2\mid X\geq1]=1+E[X^2]
$$
Wajahat
- 225
1
vote
1 answer
Game of Craps and conditional expectations
Find the conditional expected number of rolls in the game of craps given that (a)the game does not end on the first roll; (b)the player wins,but not in the first roll
The game of craps is begun by rolling an ordinary pair of dice. If the sum of dice…
Win_odd Dhamnekar
- 1,056
1
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2 answers
The order of conditional expectation
I've started this problem by defining two sigma algebras, one consisting of $F=(0, a, a^c, \Omega)$ and $G=(0,b,b^c,\Omega)$. The question I have that if I condition $E[1_a(w)\mid G]$, is $1_a(w)$ independent of $G$ then? ($1$ is indicator).
The…
Hannele
- 13
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1 answer
conditioned mean
I have a exercise, which is given as
Let the random variables $X$ and $Y$ have $f(x,y)=1$, where $-x
aa_x
- 315
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1 answer
Show that $E(E(Y|X)=E(Y)$
$X,Y$ discrete, integrable Random variables on $(\Omega,A,P)$.
Show that $E(E(Y|X))=E(Y)$.
First of all in another task it was to show that $E(Y|X)$ is a discrete Random variable. So it is
$$
E(E(Y|X))=\sum_{z}zP(E(Y|X)=z),
$$
Moreover it…
mathfemi
- 2,631
0
votes
1 answer
Conditional expecation of $\mathbf{E}[x|x>a]$
I have a very general question, and maybe stupid one. I was wondering if $\mathbf{E}[x|x>a]$ can be expressed as $1-\mathbf{E}[x|xa]$ can be written…
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0 answers
Conditional expectation inequality with squares
I am stuck on a question about conditional expectation, I was wondering if someone could give me a hint so I can make a start.
Let $X$ and $Y$ be random variables with finite mean and let $\mathbb{E}[X^2]<+ \infty$. Proof that $\mathbb{E}[X^2|Y]\geq…
dajansen
- 41
0
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2 answers
Conditional expected value of Pareto distribution
I am currently preparing for an upcoming Discrete choice models exam. I have come across an exercise where I need to calculate the conditional expected value.
Given:
$X$~$Pareto (a, b)$. Find $E(X | X > 11)$ when $(a, b) = (\frac{11}{2},…
RokasR
- 13
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1 answer
Conditional expectation on X of conditional expectation on a function of X
Sorry if the title is confusing.
I am studying this property:
$$
\mathbb{E}[\mathbb{E}(Y|W)|X] = \mathbb E[Y|X]
$$
where $X=f(W)$,
and I don't really know if I am proving it right.
Here's my proof:
$$
\begin{align}
\mathbb{E}[\mathbb{E}(Y|W)|X]
&=…
WinnieXi
- 91
0
votes
0 answers
Paradoxical formula on conditional expectation.
An accepted formula that is applied successfully in many conditional probability problems is:
$$E[f(X,Y)\mid Y=t]=E[f(X,t)\mid Y=t]$$
This formula says that the conditional expectations $E[f(X,Y)\mid Y]$ and $E[f(X,t)\mid Y]$ agree at $\quad$…
Speltzu
- 543