All Questions
Tagged with probability-theory solution-verification
65
questions
24
votes
3
answers
30k
views
Proof of the Box-Muller method
This is Exercise 2.2.2 from Achim Klenke: »Probability Theory — A Comprehensive Course«.
Exercise (Box–Muller method): Let $U$ and $V$ be independent random variables that are uniformly distributed ...
- 2,916
2
votes
1
answer
2k
views
$X$,$Y$,$Z$ mutually independent implies $X+Y$ independent of $Z$
Supposing $X$, $Y$ and $Z$ and mutually independent real random variables, how can we prove that $X+Y$ and $Z$ are independent from the definition? If not from the definition, using $\sigma$-algebras?
...
- 105
2
votes
1
answer
1k
views
Using the first and second Borel-Cantelli Lemma to find necessary and sufficient condition for convergence in probability ($98\%$ solved)
I am working an exercise with five parts, and I've solved most of them, but still have some little but nontrivial confusions. The parts (b)-(e) coincides with Durrett 1.6.15 or Durrett 2.3.15, and ...
- 5,930
3
votes
1
answer
909
views
Proof verification : $X_n \to X$ in distribution, $Y_n \to 0$ in probability $\implies$ $X_nY_n \to 0$ in probability
I have to show : $X_n \to X$ in distribution, $Y_n \to 0$ in probability $\implies$ $X_nY_n \to 0$ in probability.
Let $\alpha>0, \epsilon>0$. Then $\exists \delta>0$ such that $-\epsilon/\...
- 142
2
votes
1
answer
468
views
Law of Large Numbers for a Brownian Motion
I am self-learning introductory stochastic calculus from A first course in Stochastic Calculus by L.P.Arguin.
The part(c) of the below exercise problem on the time-inversion property of Brownian ...
- 5,450
9
votes
1
answer
5k
views
Proof of $Y=F_X(X)$ being uniformly distributed on $[0,1]$ for arbitrary continuous $F_X$
This question is related to
Showing that Y has a uniform distribution if Y=F(X) where F is the cdf of continuous X, with the difference being that $F_X$ (the probability distribution function of ...
- 3,145
6
votes
1
answer
1k
views
Two random variables are independent if all continuous and bounded transformations are uncorrelated.
Here's a statement I've come across multiple times but have never seen a proof of:
Two random variables $X$ and $Y$ are independent, if for all continuous and bounded funtions $f, g: \mathbb R\to\...
- 3,246
4
votes
0
answers
234
views
Revision of Borel-Cantelli, $(X_n)_{n \geq 1}$ sqn of $\geq 0$ identical RVs with $E(X) < \infty$ then $X_n/n \to 0$ a.s., are my arguments correct?
I care to understand the concept behind the Borel-Cantelli Lemma better, hence I would appreciate it if someone could take the time to check if my arguments below are clear and rigorous.
Statement: ...
- 3,499
3
votes
0
answers
341
views
An elegant proof of monotone convergence theorem
I have found an elegant proof of monotone convergence theorem here. I represent the proof below. It's so simple that I'm afraid if I miss some subtle detail. Could you please check if my understanding ...
- 17.6k
1
vote
1
answer
430
views
Proof of $\operatorname{E}(u(X)) = \int_{-\infty}^\infty u(x) f(x) dx$.
I want to prove that, for any monotonous function $u$:
$$\operatorname{E}(u(X)) = \int_{-\infty}^\infty u(x) f(x) dx$$
I use that $Y = u(X)$ and then use the CDF method to change variables in the ...
- 520
1
vote
1
answer
103
views
Prove that $\lvert E[X]-m\rvert \le \sqrt{\text{var}\left(X\right)}$
I want to prove that:
$$\lvert E[X]-m\rvert \le \sqrt{\text{var}\left(X\right)} $$
where $m$ is the median.
My attempt:
$$ \begin{aligned}&\lvert E[X]-m\rvert \le \sqrt{\text{var}\left(X\right)} \\...
- 4,285
1
vote
1
answer
875
views
Prove Z is a martingale by defining it is a product of random variables
Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}, \mathbb{P})$ where $\mathscr{F_n} = \mathscr{F_n}^{Z} \doteq \sigma(Z_0, Z_1, \ldots, Z_n)$, show that $Z = (Z_n)_{n \geq ...
- 13.7k
0
votes
1
answer
620
views
Prove that the Extension Theorem is Complete.
Prove that the probability triple $(\Omega, \mathcal{M}, \mathbb{P^*}) $ constructed from the Extension Theorem is complete.
Specifically, this problem asks us to show that, given any $A \subseteq \...
- 1,348
0
votes
1
answer
307
views
Find the probability of error of this channel
I am working on the following exercise:
Let $\mathcal{C} = (\mathcal{X} , P, \mathcal{Y})$ be a channel with $\mathcal{X} = \mathcal{Y} = \{0,1\}$ and transition matrix
$$ P = \begin{bmatrix}...
- 4,222
0
votes
2
answers
237
views
$\mathcal B=\sigma(\mathbb B)$ implies $f^{-1}(\mathcal B)=\sigma(f^{-1}(\mathbb B))?$
I'm having a little difficulty proving the following exercise: Suppose $(X, \mathcal A)$ and $(Y, \mathcal B)$ are two measurable spaces, and $f: X \to Y$ is measurable. Suppose also that $\mathcal B$...
- 3,145