All Questions
Tagged with probability-theory conditional-expectation
1,937
questions
0
votes
0
answers
15
views
Does is hold that $E[f(X_t)1_{\{ s \geq T_1\}}| \mathcal F_{T_1}] = P_{t-T_1}(X_{T_1})1_{s \geq T_1}$ for $s\leq t$ and X is a strong Markov process
Let $X=(X_t)_{t\geq 0}$ be a homogeneous cadlag Markov process taking values in a finite state space $S$. Let $T_1$ be its first jump time and $f$ be a bounded measurable function. I would like to ...
- 41
6
votes
1
answer
130
views
When is $\mathbb E[F(S)\mid S=s]= \mathbb E[F(s)]$ true?
Let $S$ be a discrete random variable on a set $\mathcal S$. Moreover, let $F(s)$ be another random variable (say in $L^1$) for each $s\in\mathcal S$.
Now consider some $s\in\mathcal S$. I am looking ...
- 500
1
vote
2
answers
147
views
Is such a property for conditional expectation true?
If $X$ and $Y$ are random variables defined on $(\Omega,\mathscr{F},\mathbb{P})$. Is it true that:For $\mathbb{P}$-a.s $\omega$, we have
$\mathbb{E}[X|Y](\omega) = \mathbb{E}[X|Y=Y(w)]\quad$ which $\ ...
- 123
0
votes
1
answer
67
views
Conditional probability: could we get $P(XY \ge K|{\sigma(X)}) \le \frac{X}{K}$ by $P(Y \ge K) \le \frac{1}{K}$?
Set probability space $(\Omega,\mathcal F,P)$, $\sigma$-algebra ${\mathcal V}\subset{\mathcal F} $ such that:
$X$ and $Y$ be positive random variables
for arbitrary $K > 0$, $P(Y \ge K) \le \frac{...
1
vote
0
answers
25
views
Bayes' rule for conditional expectations vs Bayes' rule for probability
I do know that there is two Bayes formulas, one for conditional expectations that is :
$$ \mathbb{E}_\mathbb{P} (X\mid \mathcal G) = \frac{\mathbb{E}_\mathbb{Q} (X L\mid \mathcal G)}{\mathbb{E}_\...
0
votes
0
answers
53
views
Definition of elementary conditional expectation through measure-theoretic conditional expectation [duplicate]
Suppose $X\in\mathbb{R}^n$ is a random vector, with given probability space $(\Omega\mathcal{F}\mathbb{P})$. The conditional probability and expectation of a random variable (for simplicity, or vector ...
- 1,022
2
votes
1
answer
48
views
A cadlag Feller process for $\mathcal F$ is Markov w.r.t $\mathcal F_+$ (Th. 46, Chap. 1, Stochastic Integration - Protter)
In page 35 of the book Stochastic Integration by P. Protter, he defines a Feller process as follows:
Then he states the following theorem.
In the proof, he used the following strategy:
Next, he ...
- 603
1
vote
0
answers
34
views
Probability theory - conditional expectation
Give an example of two random variables $X, Y$ such that $E[X|Y] = Y$ but $P(X = Y) = 0$.
I noticed that by letting $Z=X-Y$ it's equivilant to find $Z$ such that $E[Z|Y]=0$ and $P(Z=0)=0$, but I can't ...
- 2,510
1
vote
2
answers
61
views
I do an experiment which is a mixture of coin flip and uniform draw. Can I model this problem by conditional expectation as follows?
I want to model the following scenario. Suppose I flip a fair coin and if it is 'H'(Head), I draw a number uniformly from the set of real numbers $[0,M]$. So the expected value of the number I draw is ...
- 151
2
votes
1
answer
59
views
Question about notation in Durrett's book
I have been studying Markov Chains through Rick Durrett's book, more precisely I was focusing on the Markov Property (Theorem 5.2.3 on page 276 of the book available at: https://services.math.duke.edu/...
- 31
0
votes
0
answers
20
views
Critical case of Galton-Watson Process
I'm reading the proof in Artheya, Branching Process (but I think this is a classical result) of the exponential limit law in the critical case of the Galton-Watson process i.e :
$\mathcal{L}(Z_n \vert ...
0
votes
0
answers
128
views
On a conditional expectation property; substitution rule
For days now, I've been trying to prove the identity $$E(f(X,Y)\mid Y=y)=E(f(X,y)\mid Y=y).$$ I have found a couple of posts about this identity, mainly this one, and the more I think about this and ...
- 813
0
votes
0
answers
33
views
Proof of a Conditional Expectation Result Using Truncation and Conditional Jensen's Inequality
Let $X, Y$ be integrable random variables. If with probability one
$E[X|Y] = Y$ and $E[Y|X] = X$, then $X = Y$ almost surely.
(Hint: first assume $X, Y$ are $\mathbb{L}^2$, and then use truncation and ...
- 1
0
votes
1
answer
68
views
Exercise on conditional expectation
i've got an exercise that i cannot solve properly. I am given $\Omega=\mathbb{R}$, $P[A]=\int_{A}f(x)dx$ for some density function $f: \mathbb{R}\to [0,\infty)$. I am asked to give an explicit formula ...
- 21
2
votes
2
answers
61
views
Conditional Expectation of dependent normal distribution
Suppose we have $Z_0, Z_1, Z_2$ all standard normal distributed and independent. And $ X = c+aZ_0 + aZ_1$ and $Y=c+aZ_0+aZ_2$ for $a,c \ge 0$. Is there a way to calculate $E[Y|X]$ only using ...
- 615