All Questions
30
questions
3
votes
0
answers
48
views
Is my proof of Markov Property for Reflected BM correct?
I want to show that $|B_{t}|$ is a Markov Process where, $B_{t}$ is a Standard Brownian Motion. I have seen the proof here and here. But I don't understand why the method below might fail (or if it's ...
- 1,285
1
vote
0
answers
92
views
Show that: \[\mathbb{E}[X \mid Y=y]=\sum_{x \in X(\Omega)} x \mathbb{P}(X=x \mid Y=y)\]
Let $X, Y \in \mathcal{L}^{2}$ be two discrete random variables. Show that
$$\mathbb{E}[X \mid Y=y]=\sum_{x \in X(\Omega)} x \mathbb{P}(X=x \mid Y=y)$$
for all $y \in Y(\Omega)$ such that $\mathbb{P}(...
- 762
4
votes
2
answers
87
views
Conditional expectation of $X\sim U[0,2]$ given $\min(X,t)$, where $t\in [0,2]$.
I want to determine the conditional expectation of $X\sim U[0,2]$ given $\min(X,t)$, where $t\in [0,2]$. Using the definition $$\mathbb E[X|\min(X,t)] = \frac{\mathbb E[X\chi_{\min(X,t)}]}{\mathbb P(\...
- 661
4
votes
0
answers
105
views
Follow-up to "Prove $\mathbb{P}(\sup_{t \geq 0} M_t > x \mid \mathcal{F}_0)= 1 \wedge \frac{M_0}{x}$ for..."
I don't understand the accepted answer to the following question:
Let $M$ be a positive, continuous martingale that converges a.s. to zero as $t$
tends to infinity. I now want to prove that for every ...
- 3,584
2
votes
1
answer
48
views
How can I check that $\Bbb{E}(X_n1_{B_n})<\epsilon$ if $B_n=\{\Bbb{E}(X_n|F_n)\leq\epsilon\}$?
Let $X_n$ be nonegative random variables in $(\Omega, F, \Bbb{P})$ and $F_n$ a sequence of sub-$\sigma$-algebras of $F$. Let me define $A_n:=\{\Bbb{E}(X_n|F_n)>\epsilon\}$ and denote $B_n:=A_n^c$. ...
- 2,935
0
votes
1
answer
55
views
How should I interpret the conditional expectation and how to define its expression associated to each case (if it is possible)?
So I am studying the Rick Durrett book on probability theory and I am struggling to understand the notion of conditional expectation. Precisely speaking, let $(\Omega,\mathcal{F},\textbf{P})$ be a ...
- 441
3
votes
1
answer
136
views
Find $\mathbb{E}\left[X|Y=\frac{1}{4}\right]$.
Let $X,Y$ be random variables with joint density given by
$$f(x,y)=\begin{cases}
\frac{3}{8}\left ( x+y^{2} \right ) & \text{ if } 0<x<2,\text{ }0<y<1 \\
...
- 991
4
votes
0
answers
130
views
Conditional expectation given an event and a $\sigma$-algebra
Let $X$ be an integrable real random variable on the probability space $(\Omega,\mathcal A,P)$. If $A\in \mathcal A$ is an event with probability $0<P[A]<1$ then we have that
$$E[X|\sigma(A)]=...
- 4,827
3
votes
0
answers
40
views
Show that this is a random inner product
Fix a probability space $(\Omega,\mathcal A,P)$ and let $\mathcal F $ be a sub $\sigma$-algebra of $\mathcal A$. Let $\mathcal P^+$ denote the set of all random variables $X$ on $(\Omega,\mathcal A,P)$...
- 4,827
2
votes
1
answer
577
views
Is my proof of the conditionnal Jensen's inequality correct?
I want to prove the conditionnal Jensen's inequality. Let $(\Omega, \mathcal H, \mathbb P)$ be a probability space, $\mathcal G \subset \mathcal H$ a sub sigma algebra, $\varphi : \mathbb R \...
- 2,007
0
votes
1
answer
111
views
Positive conditional expectation of non-negative variable.
Consider $X \ge 0$ on $(\Omega, \mathcal{F}, P)$, suppose $P(X > 0) > 0$. Let $\tilde{\mathcal{F}} \subset \mathcal{F}$ be $\sigma$ subalgebra. Consider $Y = \mathbb{E}(X | \tilde{\mathcal{F}})$....
- 6,550
8
votes
3
answers
580
views
How can two seemingly identical conditional expectations have different values?
Background
Suppose that we are using a simplified spherical model of the Earth's surface with latitude $u \in (-\frac {\pi} 2, \frac {\pi} 2)$ and longitude $v \in (-\pi, \pi)$. Restricting attention ...
- 2,187
1
vote
0
answers
33
views
Show that this function is a conditional probability
This is an exercise from the book Probability Theory by Heinz Bauer:
Let $(\Omega,\mathcal{A},P)$ be the probability space having $\Omega:=[0,1]$, $\mathcal{A}:=\Omega \cap \mathcal{B}$, $P:=\lambda_{\...
- 4,827
0
votes
1
answer
163
views
Conditional expectation with countably many disjoint events
In his probability book Bauer gives the following prelude to the definition of conditional expectation:
Let $X$ be an integrable real random variable on a probability space $(\Omega,\mathcal{A},P)$, ...
- 4,827
1
vote
0
answers
149
views
Prove the following conditions for conditional independence are equivalent
I was asked to prove the following:
Let $X$ and $Y$ be two r.v.'s and $\mathcal A$ a $\sigma$-subfield of the probability space. $X,\ Y$ are called conditionally independent on $\mathcal A$ if for ...
- 5,261