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
0
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Conditional probability of complementary events
Let $X$ be a random variable, and consider two events $E$ s.t. $p(E) > 0$. Denote with $E^c$ the complement of $E$, and suppose that $p(E^c) > 0$. Then for any event $E'$ with $p(E') > 0$ it holds
$$
\mathbb{E} [X \mid E'] = p(E) \mathbb{E} [X \mid…
Fq00
- 1,221
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0 answers
Confusion about $\mathbb E[Y\mid X]$.
Let $\varphi :\mathbb R\to \mathbb R$ defined by $$\varphi (x)=\mathbb E[Y\mid X=x].$$
Then, $\mathbb E[Y\mid X]$ is defined by $\varphi (X)$. This definition looks a bit confusion, rigorously speaking, shouldn't it be $$\varphi (X)=\mathbb E[Y\mid…
user659895
- 1,040
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votes
1 answer
Law of iterated expectation (Binomial distribution)
I would like to know whether I applied the law of iterated expectation correctly.
$E[X|A=a]=ca$ where $X_i$ are discrete random variable ($X_i>0$).
$Y_i$ follows Binomial Distribution $Binomial(X,p)$
Then, is the following correct…
user3509199
- 175
0
votes
1 answer
Conditional expectation of a uniform distribution given a geometric distribution
Let N follow a geometric distribution with probability p. After the success of the experiment we define X, a uniform distribution from 1 to N. Both distributions are discrete. Find E[X|N].
EmKal
- 75
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votes
1 answer
Higher Order Conditional Expectation
This may be a very silly question, but does $E[y_t| I_{t-1}] \neq 0$, where $E[y_t|I_{t-1}]$ is the expectation of $y_t$ conditional on the information available at time $t-1$, implies $E[y^2_t| I_{t-1}] \neq 0$?
user27808
- 11
- 4
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If $p = f(w,a)$, what is $\mathbb{E}[w | p]$?
If $p = f(w,a)$, what is $\mathbb{E}[w | p]$?
Where $f$ is a continuous but unknown function, and $w$ and $a$ are correlated random variables.
km5041
- 105
0
votes
1 answer
Conditional Exspectation of Conditional Exspectation
Picture of the Task here
I got this task to solve and i am very dissapointet of myself that i can't solve this.
I will write only m instead of Municipality and F instead of FederlState.
For a) i got $$\mathbb{E}[wage\vert m]=2…
mastermind
- 25
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1 answer
E[X|X +Y] = E[Y|X +Y]
I am asked to prove that for two integrable independent and identical distributed random variables E[X|X +Y] = E[Y|X +Y] and then compute it .
What I have done and the way I am thinking about it is to let X+Y=Z so we can have something of the form …
Greg
- 21
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0 answers
Proof of conditional expectation example
Can someone tell me if I made any mistakes on this proof :
prove :$$E(X|A)=\frac{E[XI_a]}{P(A)}$$
where $I_a=1$ if event A occurs, else $I_a=0$.
$$E(XI_a)=I_aE(E(X|A))=I_a\sum_AE(X|A)p(A)=E(X|A)p(A)$$
With the last equality being true because…
Frank
- 880
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1 answer
For all $A \in \sigma\{X+Y\}$, we have $\mathbb{E}[X:A] = \mathbb{E}[Y:A]$?
Let $X$ and $Y$ be independent identically distributed random variables. How can one show that for all $A \in \sigma\{X+Y\}$, we have $\mathbb{E}[X:A] = \mathbb{E}[Y:A]$?
Daven
- 944
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0 answers
How to find $\mathbb{E}(V)$ where $V = \frac{100}{2^X}$ and $X\sim \Gamma(3,1)$.
How would I compute this? I tried doing:
$$
\mathbb{E}(V) = \mathbb{E}(\mathbb{E}(V|X)) = \mathbb{E} (\mathbb{E}(\frac{100}{2^X}|X=x))
$$
but that seems to not help at all... since doesn't the inner expectation just become $100/2^X$?
Mr. Bromwich I
- 574
0
votes
1 answer
Conditioning on vector in Expectation
If
$$ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} $$
and
$$ \mathbf{y} = \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} $$
Then what does
$$ E(\mathbf{y}|\mathbf{x}) = E \left( \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} \bigg| \begin{bmatrix} x_1…
Sunhwa
- 387
0
votes
1 answer
Conditional expectation for exponential density
I want to find what is $\mathbb{E}(X|Y)$, where X follows Exp(1) and $Y:=\left \lfloor X \right \rfloor$.
My idea:
First, $\mathbb{P}(Y=n)=\mathbb{P}(X \in [n,(n+1)))=\int_{n}^{n+1}e^{-x}dx=e^{-n}(1-1/e).$
Second,…
Melina
- 937
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votes
1 answer
Conditional expectation - example based on definition
$$
\Omega=\{a,b,c\}, \mathcal{F}=2^{\Omega}\\
P(\{a\})=P(\{b\})=P(\{c\})=\frac{1}{3}\\
X(a)=1, X(b)=2, X(c)=15
$$
$$\mathcal{G}=\sigma (\{a\})=\{\phi, \{a\}, \{a,b,c\}, \{b,c\}\}\}$$
I have to find $E(X|\mathcal{G})$.
I have a…
michelson
- 109
0
votes
1 answer
Conditional Expectation in a uniform distribution
Q:Pick a point $(X, Y )$ inside the unit square $[0, 1]^2$ uniformly at random. Let $Z = X^2+Y^2$
What is $E[X|Z]$?
A: I believe i get the answer to be $E[X|X^2] +1/2$,
but i don't know how to calculate $E[X|X^2].$
Any help?
