0
votes
0answers
29 views

Product of stochastically independent random variables

Let $X, Y, Z$ be three stochastically independent random variables that are quadratic integrable (quadratintegriertbar is the German term, I didn't find a exact translation). No which statements hold ...
2
votes
1answer
70 views

Product of $n$ i.i.d. random variables

Let the variable $Z$ equal $Z = XY$ where $X$ and $X$ are two i.i.d. continuous random variables which distributions are given by $f_X()$ and $f_Y$. The distribution of $Z$ is given by: $$f_Z(z) = ...
2
votes
0answers
87 views

Absolute Continuity and simple discontinuity

I am reading a book called Stochastic Process, Estimation, and Control, in P.32 it states that a function with finite simple discontinuities can still be absolutely continuous, which confused me, I ...
2
votes
3answers
224 views

Proof of Levy's zero-one law

Let $(\Omega, \mathcal{F},\mathbb P)$ be a probability space and let $X$ be a random variable in $L^1$. Let $(\mathcal{F}_k)_k$ be any filtration, and define $\mathcal{F}_{\infty}$ to be the minimal ...
1
vote
2answers
64 views

$X_1,X_2,…$ real independent random variables $\not\implies S_n = \frac{1}{n}(X_1+ \cdots + X_n)$ are independent

Is it possible to show the following Lemma: Given independent real random variables $(X_i)_{i\in \mathbb{N}}$, then $(S_n := \frac{1}{n}(X_1+\cdots + X_n))_{n \in \mathbb{N}}$ are independent as ...
1
vote
1answer
510 views

Joint density $X^2+Y^2$

Let's say we have a point $(x,y)$ in the unit circle. I've read (without proof :( ) that the joint density of $z$, where $z^2=x^2+y^2$, is: $$f_{X,Y}(x,y) = ...
0
votes
0answers
41 views

The identity of two parameters derived via conditioning arguments

Suppose I have a random variable $X_1\in\mathbb{R}$ and a random vector $X_2\in\mathbb{R}^d$. Furthermore, there are two measurable functions $f_1$ and $f_2$, and two deterministic vectors $\theta_1, ...
2
votes
1answer
331 views

Partial derivative respect to random variable - How does one compute this?

CLARIFICATION: If someone could please help me understand the following: When examining the expected value in this specific situation, how is the distribution of $\theta$ relevant? What ...
1
vote
0answers
167 views

Independent Exponentially Distributed Random Variables - Athletes Problem??

Q) At a javalin competition two athletes (1 & 2) are competing against each other. Each has one attempt to throw the javalin. Assume the acheived distance of a throw ($L$1 & $L2$) [note these ...