All Questions
135
questions
3
votes
0
answers
77
views
Correlation of conditional expectation of uncorrelated random variables
Let $X,Y\in\mathcal{L}_{2}\left(\Omega,\mathcal{F},\mathbb{P}\right)$ satisfy $\mathbb{E}\left[X\right]=\mathbb{E}\left[Y\right]=\mathbb{E}\left[XY\right]=0$, and $\mathbb{E}\left[X^2\right]=\mathbb{E}...
- 191
0
votes
2
answers
54
views
Does expectation inequality imply conditional expectation inequality?
Given a probability space $\left(\Omega\text{, }\mathcal{F}\text{, }\mathbb{P}\right)$ and two random variables defined on it, does it hold true that
$$
\mathbb{E}\left(X\right)<\mathbb{E}\left(Y\...
0
votes
1
answer
41
views
Is it true that $\mathbb{E}\{X\}=\mathbb{E}\{Y\}$ $\Rightarrow$ $\mathbb{E}\{X|\mathcal{F}\}=\mathbb{E}\{Y|\mathcal{F}\}$? [closed]
Given $(\Omega$, $\mathcal{F}$, $\mathbb{P})$ and two r.v.'s $X$ and $Y$ defined on it, does it hold true that:
$$\mathbb{E}\{X\}=\mathbb{E}\{Y\}\Rightarrow\mathbb{E}\{X|\mathcal{F}\}=\mathbb{E}\{Y|\...
- 1
1
vote
1
answer
277
views
Conditional expectation for independent random variables
I have a question about the conditional expectation with some independence conditions for random variables and $\sigma$-fields.
For a random variable $X$ with $E|X| < \infty $, if $Y_1$ and $ ...
- 27
4
votes
1
answer
224
views
Law of large numbers holding uniformly with respect to a distribution
Let $X$ and $\varepsilon$ be independent random vectors, $\mathcal{X} = \text{supp}(X)$, and $Y = f(X) + \varepsilon$ for some function $f$.
For any $x \in \mathcal{X}$, let $y^i = y^i(\omega)$, $i \...
- 1,788
1
vote
2
answers
154
views
Assumption of almost supermartingagles convergence theorem
My question is about the assumption of the following result, which is due to Robbins and Siegmund.
Suppose that $(\varOmega,\mathscr{F},P)$ is a probability space.
Let $(\mathscr{F}_n)_{n\in\...
- 1,009
3
votes
1
answer
103
views
Have I incorrectly used conditional expectation here?
Let $N$ be a Poisson random variable with parameter $\lambda$. If the parameter $\lambda$ is not fixed, but an exponential random variable with parameter $1$, find $E[N]$.
Here is a correct solution ...
- 1,503
2
votes
1
answer
60
views
"co-relatedness" of conditional expectation of two independent random variables
Let $\mathcal{F}$ be a $\sigma$-algebra and $X,Y$ be two independent (not just uncorrelated) random variables, I wonder if the following statement true
$$\mathbb E(XY|\mathcal{F})=\mathbb E(X|\...
- 8,039
2
votes
1
answer
262
views
Is it feasible to prove this property of conditional expectation without too many lemmas?
I'm reading a proposition about conditional expectation operator in the lecture note:
My lecture note does not provide the proof. I would like to ask it's possible to prove it from below ...
- 17.6k
1
vote
0
answers
383
views
Conditional Expectation Tower Property proof exercise
In the answer, i'm not sure how do derive the density of Y given Z. Also, why are we integrating with respect to y but not to z? I know by definition that if we're searching for expectation of Y given ...
user634512
0
votes
1
answer
42
views
Hints to calculate Show $E[X_{1}\vert Y]=\frac{1}{5}(Y-1)$
We roll a fair dice until we reach roll a $6$. Let $Y$ be an RV representing the rolls needed to roll a $6$ and $X_{1}$ be the number of throws wherein we threw a $1$.Show $E[X_{1}\vert Y]=\frac{1}{5}(...
- 1,838
1
vote
1
answer
104
views
Using conditional expectation with MSE function
I'm trying to understand a derivation step in applying conditional expectation to the following starting function:
$f_* = E_{P_{(x,y)}} [(y-f(x))^{2})]$
and how, after applying conditional ...
- 167
1
vote
2
answers
59
views
Question about conditional expectation and sum of random varibles
Suppose that $X,Y,Z$ are random variables and $f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is measurable.
I think that the following is true
$$\mathbb{E}[\mathbb{E}[f(X+Y,Z)|X,Y]X]=\mathbb{E}[\...
- 271
4
votes
1
answer
1k
views
Proving $V(X) = E(V(X|Y)) + V(E(X|Y))$ using the pythagorean theorem
I know the textbook proof of $$V(X) = E(V(X|Y)) + V(E(X|Y))$$ but I'm interested in understanding the weird proof/analogy with the pythagorean theorem my professor gave in class.
With $X, Y$ random ...
- 946
2
votes
1
answer
2k
views
Finding the conditional expectation of independent exponential random variables
Let $X$ and $Y$ be independent exponential random variables with respective rates $\lambda$ and $\mu$. Let $M = \text{min}(X,Y)$. Find
(a) $E(MX|M=X)$
(b) $E(MX|M=Y)$
(c) Cov$(X,M)$
(a) I first ...
