All Questions
135
questions
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
1
answer
136
views
How does one compute conditional expectation with respect to a continuous random variable?
I can't wrap my head around the way to compute conditional expectation with respect to a continuous random variable. For instance, consider a probability space $(\Omega, A, P)$, where $\Omega = [0,1]$ ...
- 263
1
vote
1
answer
343
views
Proving conditional expectation is bounded if expectation is bounded
Suppose we have a random variable $X$ bounded in expectation, i.e., $E[X] < c$.
If $X \geq 0$, then is it correct to infer that the conditional expectation of $X$ given another r.v. $Y=y$ is ...
- 219
0
votes
1
answer
128
views
Conditional Expectation of Function of I.I.D Random Variables [duplicate]
Let X,Y,Z $\stackrel{i.i.d}{\sim}$ N($0$,$1$). I am supposed to find the E($2$X+$3$Y | X+$3$Y-Z =$4$)
I Tried to solve the problem by considering A= $2X+3Y$ ~ $N(5,5)$ and B=X+$3$Y-Z ~ $N(3,5)$ but ...
- 119
2
votes
3
answers
263
views
What is the distribution of $E[X\mid Y]$?
Let $(X, Y)$ be two r.v. with joint p.m.f. described by the following table
What is the marginal distribution of $X$?
Are $X$ and $Y$ independent?
What is the conditional p.m.f. of $X$ given $Y=0$?
...
- 1,583
10
votes
1
answer
392
views
What am I writing when I write $\mathbf X \mid \mathbf Y$?
Suppose $\mathbf X$ is a random variable and $A$ is an event in the same probability space $(\Omega, \mathcal F, \Pr)$. (Formally, $\mathbf X$ is a function on $\Omega$, say $\Omega \to \mathbb R$; $A$...
- 145k
0
votes
0
answers
145
views
Range of conditional expectation
Let $X$ be a random variable with values in $A$.
Let $Y$ be random variable with values in $B$.
Let $g: A\times\mathbb R \to C$ a differentiable function.
Define $h(z)= \mathbb E(g(X,z)\mid Y=y)$.
...
- 73
1
vote
2
answers
74
views
What is the expectation of $(\langle x, w \rangle - \langle y, w \rangle)^2$, where $x,y,w$ are independent Bernoulli random vectors?
I'm stuck on the following problem.
Consider three independent Bernoulli random vectors $x,y,w$ of length $n$, where each entry follows the Bernoulli distribution $B$ with $P(B=0)=P(B=1)=\frac{1}{2}$.
...
- 37
1
vote
1
answer
113
views
$X=\mathbb{E}[X\mid\mathcal{F}] $ implying $\mathcal{F}$-measurability of $X$
For a random variable $X$ we know, that if $X$ is measurable w.r.t some filtration $\mathcal{F}$ it satisfies $X=\mathbb{E}[X\mid\mathcal{F}]$.
I wonder whether one can reverse this property and state,...
- 543
3
votes
1
answer
159
views
Effect on expected value of conditioning on inequality between random variables (do we have E[X | X>S] ≥ E[X | X>S, Y>S]?)
I've been trying to prove the following inequality:
$$
\mathbb{E}[X \mid X > S] \geq \mathbb{E}[X \mid X>S, Y>S]
$$
where $X$, $Y$, $S$ are mutually independent real-valued random variables ...
- 31
1
vote
1
answer
957
views
Proof Linearity of Conditional expectation
How we can proof that:
$E[X - Y|W] = E[X|W]-E[Y|W]$
I try to use the definition of Conditional Expectation:
$E[X|Y=y]= \sum_x \cdot p(x|y)$ and then to substitute $X=A-B$
Is it the right way? And ...
user840425
11
votes
1
answer
1k
views
Does almost sure convergence and $L^1$-convergence imply almost sure convergence of the conditional expectation?
Question. Let $ X_{n}, X $ be random variables on some probability space $ ( \Omega, \mathcal{F},\mathbb{P} ) $ and let $ \mathcal{G} \subset \mathcal{F} $ be a sub-$\sigma$-algebra. Moreover ...
- 115
4
votes
1
answer
320
views
Approximating $E[g(Y)]$ with a Taylor series is easy, but $E[g(YX)]$ seems much tougher (potentially impossible)?
Let $X$ and $Y$ be continuous random variables with $Y \sim \mathcal{N}(\mu_Y,\sigma_Y^2)$.
I want to approximate $E[g(YX)]$ where $g(x) = e^{-x^2}$.
Specifically, my plan is to use a Taylor expansion ...
- 10k
2
votes
3
answers
256
views
How to show $E[X\mid X = x] =x$?
I read that $E[X\mid X = x] =x$ but I don't get that when I try to prove it:
\begin{align}
E[X\mid X = x] &= \sum x P(X=x|X=x) \\
&= \sum x \frac{P(X=x,X=x)}{P(X=x)} \\
&= \sum x \frac{P(X=...
- 165
0
votes
1
answer
27
views
Question about conditioning on sequences of random variables, where the sequence is related by a function $f$
Let $A_k, B_k$ be two sequences of random variables, where $A_k = f(B_k)$
Then is it true that
$$\mathbb{E}[A_{k+1}|A_0, \ldots, A_k] = \mathbb{E}[f(B_{k+1})|B_0, \ldots, B_k]$$
If so, what is the ...
- 557