Questions about maps from a probability space to a measure space which are measurable.

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4
votes
0answers
105 views

Sigma algebra generated by a homeomorphic random variable

Let $\Omega = [0,1]$ be our probability space with sigma algebra of borel sets on $[0,1]$ and Lebesgue measure on $[0,1]$. Let Y be a random variable such that $Y(\omega) = Y(1-\omega)$ for every ...
0
votes
0answers
16 views

Modifying a generator of random numbers from a trapezoidal distribution to include growth and decay rates

I've written a C# random number generator based on page 11 of this paper: http://pubs.usgs.gov/tm/04/c03/tm4-C3_final_508_files/tm4-C3_apdx1_v030813.pdf It works fine but I would like to modify it, ...
1
vote
0answers
205 views

Conditional expectation, quadratic function, absolute value

We are given two random variables defined on $[0,1]$: $$X(\omega) = 2 \omega -1 + |2 \omega -1|$$ $$Y(\omega) = 1-|2 \omega^2 -1|$$ I am supposed to find $\mathbb{E}(X|Y)$ which by definition is a ...
2
votes
2answers
44 views

is there a concept of asymptotically independent random variables variables?

To prove some results using a standard theorem I need my random variables to be i.i.d. However, my random variables are discrete uniforms emerging from a rank statistics, i.e. not independent: for ...
0
votes
0answers
36 views

independence copula diagonal

I'm reading Nelsen's Instruduction to copulas, and there is (probably very simple) excersice I cannot deal with. It says that if the diagonal section of the copula equals the diagonal of independence ...
1
vote
1answer
34 views

Dice roll, estimator, epsilon

We roll a non-symmetric die. Let $X_n$ be the reulst of $n$-th roll. $$P(X_n = 6)= \frac{1}{6} + \varepsilon, \ P(X_n = 1) = \frac{1}{6} - \varepsilon, \ P(X_n=2) = ... = P(X_n = 5) = \frac{1}{6} $$ ...
4
votes
4answers
258 views

Show the probability that the sum of these numbers is odd is 1/2

Setting Let $S$ be a set of integers where at least one of the integers is odd. Suppose we pick a random subset $T$ of $S$ by including each element of $S$ independently with probability $1/2$, Show ...
0
votes
1answer
13 views

Distribution of sine composed with a random variable

Could you tell me if my calculations are correct? We are given a random variable with the following discrete distribution $$P(X=n) = \frac{2^n}{3^{n+1}}, \ \ n \in \mathbb{N}.$$ Find the ...
0
votes
1answer
31 views

Example random variable $\xi$ such that $\xi$ and $\xi^2$ are independent

Find random variable $\xi$ such that $\xi$ and $\xi^2$ are independent.
1
vote
1answer
26 views

cdfs $F$ and $G$ of random variable $X$, $F\le G$. What can we say about $\mathbb{E}_F[X]$ and $\mathbb{E}_G[X]$?

Problem: A random variable $X$ is distributed in $[0,1]$. Mr. Fox believes that $X$ follows a distribution with cumulative density function $F:[0,1]\to [0,1]$ and Mr. Goat believes that $X$ follows a ...
0
votes
1answer
19 views

$X$ and $Y$ are unformly distributed in $[0,1]$ with $P(\max(X,Y)≤z)=P(\min(X,Y)≤(1−z))$. Find $z$.

Problem: Two independent random variables $X$ and $Y$ are uniformly distributed in the interval $[0,1]$. For a $z \in [0,1]$, we are told that $P(\max(X,Y)\le z)=P(\min(X,Y)\le (1-z))$. Then, what is ...
1
vote
1answer
23 views

Inheritance of independence of random variables

I want to show the following statement: let $(X_n)$ and $(Y_n)$ be sequences of random variables and $X_n\perp Y_n$ for each $n$. If $X_n\to X$ and $Y_n\to Y$ in probability respectively, then $X\perp ...
1
vote
1answer
21 views

If a sequence of random variables all have the same mean, is the sequence tight?

Suppose $(X_n)$ are almost surely non-negative random variables all with the same finite mean $\mu$. Is this sequence necessarily tight?
1
vote
2answers
48 views

Sigma algebra generated by an absolute value random variable

I need to find out what the sigma algebra generated by $Y$ looks like for $$Y: [0,1] \ni (\omega) \to 1- |2\omega -1| \in \mathbb{R}.$$ The graph of $Y$ is symmetric with respect to $\omega = ...
1
vote
1answer
19 views

Meaning of the random variable Y=|X|

I am learning this and having a very basic doubt. Suppose $X$ and $Y$ are two random variables where $X$ takes the values $-2,-1,0,1,2$ each with probability $1/5$ and $Y=|X|$. I think $Y=|X|$ means ...
0
votes
1answer
24 views

Distribution of Logistic of Normal

If $X \sim N(\mu_X, \sigma^2_X)$ and $Y= \frac{\exp(X)}{1+\exp(X)} $, what is the distribution of $Y$? I thought logit-normal would fit the bill, however the logit of $Y$ is ...
1
vote
0answers
13 views

Do derivative and expectation operators commute only for discrete random variablesÉ

Suppose we have a function $$g(r)=E(f(r,X))$$ where $X$ is a random variable and $g:\mathbb{R}\rightarrow\mathbb{R}$. If $X$ is a discrete random variable, we can simply write $$g(r)=\sum_{x\in ...
0
votes
0answers
32 views

conditional expectation given two conditions

I want to check my understanding of conditional expectation. Could someone confirm if this is true? Y(t) is normally distributed. E[Y(2)|Y(1),Y(3)] = E[Y(2)|Y(1)] + E[Y(2)|Y(3)] If this is not ...
2
votes
1answer
42 views

An equivalent condition for a random variable to be integrable

I have to prove the following fact. Show that $X_1$ is integrable, iff for all $\epsilon>0$ $$\sum_{n=1}^{\infty} \mathbb{P}(|X_1|>n \epsilon)<\infty.$$ Here $X_1$ is just a random ...
0
votes
1answer
18 views

Issue with sum of probabilities of probability distribution function of a geometric random variable

Is it possible that the sum of probabilities of geometric distribution for "$k = 1,...,n$", where k is number of trials until the first success, is not equal to 1? I'm asking this, because I encounter ...
0
votes
0answers
35 views

Independent random variables and integrability

This is a problem that I am stuck at. I think I have to prove the hint first. But I can't find a way to prove the 'only if' part of the hint. (the 'if' part is just manifest). Could anyone help me ...
0
votes
0answers
12 views

signal variance

I read a section about $H_2$ control. And one section is as following: It seems that the context assumes 1. $w(t)$ is a vector. i.e. $w(t) = [w_1(t) ...w_n(t)]^T$ 2. Each entry is zero mean. ...
0
votes
4answers
40 views

Can the range of a variable be inclusive infinity?

Can a range be $[0, \infty]$ or must it be $[0, \infty)$ because you can never quite reach infinity? Clarification: $[0, 1]$ means $0 \leqslant x \leqslant 1 $, while $(0, 1)$ means $0 < x < ...
2
votes
1answer
37 views

Which matrices are covariances matrices?

Let $V$ be a matrix. What conditions should we require so that we can find a random vector $X = (X_1, \dots, X_n)$ so that $V = Var(X)$? Of course necessary conditions are: All the elements on ...
1
vote
2answers
37 views

What does $\sim$ in $X\sim \mathcal{N}(\mu,\sigma^{2})$ really mean?

This is a bit of a silly question, but I can't seem to find the answer anywhere. I feel like $X\sim \mathcal{N}(\mu,\sigma^{2})$ means that $\sim$ is a relation, but if it is a relation, what ...
1
vote
1answer
19 views

Finding a formula for a probability density function

The 75th percentile of a random variable X is the value X=k such that 75% of the observed values of X are less than k. For example, if the 75th percentile on an exam is 87, then 75% of the scores are ...
5
votes
2answers
128 views

Proving the sum of two independent Cauchy Random Variables is Cauchy

Is there any method to show that the sum of two independent Cauchy random variables is Cauchy? I know that it can be derived using Characteristic Functions, but the point is, I have not yet learnt ...
2
votes
2answers
42 views

how to understand the generation of cauchy distribution from uniform distribution?

I am learning some basic idea on generating cauchy distribution from uniform random generator $u \in [0, 1]$. I know it was discussed before in How to generate a Cauchy random variable, but during my ...
1
vote
1answer
29 views

Characteristic Function limit to 0

When calculating the limit of the following characteristic function $$ \frac{1}{n+1}\left[ \frac{1-\exp\left( \left(n+2 \right)it \right)}{ 1-\exp(it) } \right]$$ and taking its limit when ...
0
votes
0answers
15 views

Closed form expression for the S-transform of a random variable?

I'm trying to compute the S-transform as described in this review article on random matrix theory (section 2.2.6). They define it as $\Sigma_X(\gamma) = -\frac{\gamma+1}{\gamma}\eta_X^{-1}(1+\gamma)$ ...
-2
votes
1answer
39 views

Proving the variance of pareto random variable equals (a*lambda)/((a-1)^2*(a-2))

So my PDF for the Pareto distribution is: $$\dfrac{a\lambda^a}{x^{a+1}},\quad x\ge\lambda$$ To find the variance, you need to find the integral of $x^2\dfrac{a\lambda^a}{x^{a+1}}$ and subtract it from ...
1
vote
2answers
40 views

Solution to a covariance problem

Roll two dice, let $X$ be their sum, and $Y$ be the second roll subtracted from the first. Compute $\text{Cov}[X,Y]$. Brute-force calculating $\text{Cov}[X,Y]=E[XY]-E[X]E[Y]$ is fairly horrible, ...
0
votes
1answer
9 views

Proof that 2 geometric random variable is NB

can someone write me the proof of 2 geometric variable are negative binomial ? $X\sim G(p)$ and $Y\sim G(p)$ how can i proof that $Z=X+Y \sim NB(2,p)$?
2
votes
1answer
64 views

Conditional expectation, $X = \varphi (Y)$

Show that if $$\forall \omega \in A \ : \ X(\omega) = \varphi(Y(\omega)), \ \ A \in \Sigma_Y$$ (that is, the equality is true for $\omega \in A$), then $$\mathbb{E}(X|Y)(\omega) = \varphi(Y(\omega)) ...
1
vote
2answers
57 views

How to calculate the probability distribution function (PDF)?

Sorry for the dumb question, I've been struggling with understanding the probability distribution function formula, what does "x" and "d" stand for in the formula , and how to use the formula? I've ...
0
votes
0answers
8 views

how can find statistical and linear independence from sample space

Let sample space is $ S=\{\varsigma_1,\varsigma_2,\varsigma_3,\varsigma_4\} =\{-1, -0.5,0.5,1\}$ and define two random variables as $ X(\varsigma)=1/\varsigma $ and $Y(\varsigma)=2^{-\varsigma}$. a) ...
6
votes
1answer
43 views

Divergent series of random variables

I've been trying to prove that given a sequence of independent random variables with identical distribution $\{X_n\}_{n \in \mathbb{N}}$ such that $P(X_1 \neq 0)>0$, so also $P(X_i \neq 0) >0 \ ...
1
vote
3answers
48 views

How to explain why the probability of a continuous random variable at a specific value is 0?

Consider X as a continuous random variable which can assume any value in [0, 1]. It is known that P(X=x)=0 where P is the probability density function. I want to understand this intuitively. The math ...
0
votes
1answer
41 views

2 User Queuing Model Probability Problem

Consider two users who arrive to a system with exponential arrival times with parameters $\lambda_a$ and $\lambda_b$. Once they arrive, the users stay in the system for an exponentially distributed ...
0
votes
4answers
98 views

Expected value of the minimum of a non-negative random variable and a constant

X is a non-negative random variable. Define Y = MIN(X, c) where c is a constant. What is E[Y]? I am modeling the constant as another random variable whose pdf is Dirac Delta function: $f_{c}(x) := ...
2
votes
1answer
69 views

Sequence of finite, positive and i.i.d random variables and limit of $\frac{S_{n+1}}{S_{n}}$

Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence of finite, positive and i.i.d random variables and let's call $\mu:=E(X_{1})>0$ and $S_{n}:=\sum_{i=1}^{n}X_{i}$. We know that ...
0
votes
2answers
48 views

CDF of $-\ln X$ where $X$ is uniform on $(0,1)$

I'm having difficulty studying this part of the subject, because i can't get through this first example, can anyone help? Let $$X: U(0,1)$$ Find the distribution function of the following random ...
0
votes
0answers
20 views

In a group of 2n couples , in a random fashion 2n people are chosen.

In a group of 2n couples , in a random fashion 2n people are chosen. Find the distribution of the random variable X , which represents the number of couples in the chosen group. $\Omega$-probability ...
1
vote
1answer
31 views

Let $X$ and $Y$ be two uniformly distributed random variables on $[0,1]$. Find $E(X^k)$ and $E(XY^k)$.

Let $X$ and $Y$ be two uniformly distributed random variables on $[0,1]$. Find $E(X^k)$ and $E(XY^k)$. How can you do this? Do you need the change of variables technique? I am a bit confused about ...
1
vote
1answer
26 views

continuous probability density functions

Continuous distributions assign probability 0 to individual values. But, according to DeGroot, it doesn't mean that it is impossible for the random variable to take individual values. So, why not make ...
2
votes
2answers
64 views

Expectation of quotient of random variables

Let $X_1,...X_n$ be independent, identically and non negative random variables, and let $k\le n$. Compute: $$E\left[{\sum_{i=1}^k X_i\over \sum_{i=1}^n X_i}\right]$$ this question has already been ...
2
votes
1answer
27 views

Random number generator from a piecewise PDF

I'm trying to create a random number generator on the interval $(a,c)$ given a probability density function defined as: $$f(x) = \left\{ \begin{array}{lr} \dfrac{C}{x} &, x \in (a,b)\\ ...
2
votes
2answers
51 views

Is there a meaningful way to approximate a discrete random variable?

Is there a meaningful way to find a continuos approximation of a discrete random variable? Thoughts for the $L^2$ case If $X \in L^2$, then we may want to consider the subspace $V = C^1 \cap L^2$ ...
0
votes
1answer
34 views

Would it be safe to say that a random variable X is identially zero when its first and second moments are both zero?

Would it be safe to say that a random variable $X$ is identially zero when its first and second moments are both zero? If it is true, how would you prove this? This step is needed when we prove that ...
1
vote
1answer
46 views

How do we approximate sum of random variables?

Suppose we have independent, identically distributed random variables $X_n \notin L^1$. I would like to approximate, in some way, the distribution of their sum $\sum X_n$ .The problem is that these ...