All Questions
137
questions
-1
votes
1
answer
36
views
Sigma-algebra generated by $\sin(2\omega)$
If $\Omega = (0, 2\pi)$ and $X(\omega) = \sin(2\omega)$, is $\sigma(X) = \{S: x \in S \cap [0, \pi / 4] \Leftrightarrow \pi / 2 - x \in S \cap [\pi / 4, \pi / 2] \Leftrightarrow \pi + x \in S \cap [\...
2
votes
1
answer
46
views
Sufficient Condition on Almost Surely Convergence
Let $f_n \in [0, 1]$ and suppose if we want to show
$$
\lim_{n \to \infty} f_n = 1
$$
almost surely, is it enough to show
$$
\lim_{n \to \infty} \mathbb{P}\{ f_n = 1 \} = 1?
$$
If not, what if we add ...
- 593
0
votes
1
answer
42
views
$X$ a r.v. verifying $P(X=x)=0$&$F_X$ its repartition fct. Prove that $F_X(X)$ follows an uniform distribution law $]0;1[$.
Question:
Let $X$ a r.v. verifying $P(X=x)=0, \forall x \in \mathbb{R}$ and $F_X$ its repartition function. Prove that $F_X(X)$ follows an uniform distribution law $]0;1[$.
Answer:
1- We write $0 \leq ...
- 719
0
votes
1
answer
40
views
If $\sigma$-algebra generated by $X$ is a subset of $\sigma$-algebra generated by $X^2$.
The question is about $\sigma$-algebras and to prove or give a counter-example to the statement that for $X$ a random variable, $\sigma(X) \subseteq \sigma(X^2)$.
I think the statement is false. ...
- 367
0
votes
1
answer
78
views
Let $(A_k)_{k \in \mathbb{N}}$ a sequence of events.If $P(A_k)$ does not converge to $0$ then $\exists$ an event belonging to an infinity of $A_k$
Question:
Let $(A_k)_{k \in \mathbb{N}}$ a sequence of events. Prove that if $P(A_k)$ does not converge to $0$ then $\exists$ an event belonging to an infinity of $A_k$
My answer:
1-Let writte $A'_1=...
- 719
1
vote
1
answer
48
views
Convergence of random variables $X_{n}=e^{n\alpha}1_{[n,\infty)}$ in specific probability space proof
Suppose $(\Omega, F, P)$ is probability space, where $\Omega =[0, \infty)$, $F$ is Borel $\sigma-$ algebra on $[0, \infty)$ and $P$ probability $P(B)=\int_{B}^{} e^{-x} \,dx, \forall B \in F$. Let's $(...
- 97
2
votes
0
answers
69
views
Proving $\mathrm E[XY] = E[X] E[Y]$ whenever $X$ and $Y$ are independent random variables.
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability measure space. Let $X$ and $Y$ be two independent random variables on it. Then $\mathrm E [XY] = \mathrm E [X] \mathrm E [Y].$
I tried to do it ...
- 2,612
0
votes
0
answers
29
views
Proof that $\mathcal{A}_1 \cap \mathcal{A}_2$ forms a valid $\sigma$-algebra over $\Omega$ using Boolean algebra
I'm trying to proof that $\mathcal{A}_1 \cap \mathcal{A}_2$ forms a valid $\sigma$-algebra over $\Omega$, if $\mathcal{A}_1$ and $\mathcal{A}_2$ are both each a valid $\sigma$-algebra over $\Omega$, ...
- 145
2
votes
1
answer
103
views
Show that $\mathbb{E}(\mathbf{1}_{\{X>t\}}|\mathcal{G})$ is $(\mathcal{B}(\mathbb{R})\otimes\mathcal{G},\mathcal{B}(\mathbb{R}))$-measurable.
Let $X$ be a non-negative integral random vairable on $(\Omega,\mathcal{F},\mathbb{P})$ and let $\mathcal{G}\subseteq\mathcal{F}$ be a sub-$\sigma$-algebra. I want to show that $$\mathbb{E}(X|\mathcal{...
- 5,930
0
votes
2
answers
182
views
Sufficient condition for $L^1$ convergence using uniformly integrabllity
I would like to prove the following result : let $(Xn)_n$ be a sequence in $L^{1}(\Omega,\mathcal{F}_t, \mathbb{P})$ that converges almost surely to $X\in L^1$. Then if $(X_n)_n$ is uniformly ...
- 1,381
-1
votes
1
answer
67
views
Prove that random variable is a stopping time
After a question I asked concerning my failure to prove that a random variable is a stopping time I come to propose another proof that I hope is better.
We consider $X$ a continuous process with its ...
- 1,381
2
votes
0
answers
80
views
$0$-$1$ law for brownian motion
I would like to prove the following theorem : Let $B$ be a brownian motion and $\mathcal{F_t}$ its natural filtration. Then for all $A\in\mathcal{F}_{0^{+}}$ we have $\mathbb{P}(A)\in\left\{0,1\right\}...
- 1,381
0
votes
0
answers
66
views
Modification of a stochastic process and complete filtration
I consider $B_t$ $t\in[0,T]$ a (real valued) stochastic process adapted for the filtration $\mathcal{F}_t$ and $\bar{B}_t$ à modification of $B_t$. I would like to show that $\bar{B}_t$ is adapted ...
- 1,381
1
vote
0
answers
33
views
Probability Measure and Algebraic Structure in Infinitely Repeated Coin Tosses
Hey I want to check my solutions for this problem:
Consider the sample space $\Omega = \{0, 1\}^N$ of an infinitely repeated coin toss. Let $Π_n: \Omega \rightarrow \{0, 1\}^n$ be the coordinate ...
- 900
1
vote
0
answers
36
views
Relationship Between $\sigma$-Continuity and Probability Measures in Set Theory
I want to check my solutions for this problem:
Let $F$ be a $\sigma$-algebra over $\Omega$, and $Q: F \rightarrow [0, 1]$ be a normalized, additive set function (i.e., $Q(\Omega) = 1$ and $Q(A \cup B) ...
- 900