Questions tagged [random-variables]

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

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28 views

Are “independent events” in probability really independent? [closed]

This is a hard and deep question. I understand very well the concept of independence. But, let us take two events: Event A (I throw a dice) and event B (some star explodes in an near galaxy). Are ...
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11 views

Find the limit cdf for a sequence of random varibales whose characteristic functions do not converge.

X is a random variable with $P(x=-1)=P(x=1)=\frac{1}{2}$, $Y_n=nX$ is a sequence of random variables, n=1,2, $\dots$ $S_n = \frac{1}{n}\sum_{i=1}^{n} Y_i$ is the partial sum of $Y_n$. I am asked to ...
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42 views

Problem with module

Knowing that… $Y:=\left\{\begin{matrix} X & if & |X|<a\\ -X & if & |X| \geq a \end{matrix}\right.$; $X \sim N(0,1)$; $a>0$; …and obviously $|X|:=\left\{\begin{matrix} X & ...
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1answer
21 views

General form of the open mapping theorem

Let $X,X_1,X_2,...$ be real valued random variables on the same probability space $(\Omega, \mathcal{F},\mathbb{P})$. Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function.We know that ...
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1answer
18 views

Does the Martingale Representation theorem hold both ways?

Can the Martingale Representation theorem be used to assume that the integral with respect to Brownian motion, $B(t,\omega)$, $$X=\int^{T}_{0}B^{4}(t,\omega)dB(t,\omega)$$ is a square integrable ...
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1answer
20 views

Density of $g(Y)=\frac{1}{2}\mathbb{E}[X|Y]$

Let $(X,Y)$ a random variable with density $f(x,y)=cx(y-x)e^{-y}$ for $0 \leq x \leq y <\infty$. Find: 1) the value of $c$. $\rightarrow c=1$ 2) the density of $X|Y=y$. $\rightarrow f_{X|Y}(x,y)...
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38 views

What's the expectation of the function given by $|X_1|$ when $X=(X_1 \dots X_d) \sim Unif(S^{d-1}(r))$

Let $X=(X_1 \dots X_d) \sim Unif(r S^{d-1} \equiv S^{d-1}(r) ).$ I'd like to know what's the expectation of $|X_1|,$ i.e. what's the integral: $$ \frac{1}{vol(S^{d-1}(r))} \int_{S^{d-1}(r)} |x_1| d ...
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14 views

Show the convergence of $\sum_{i=1}^{n}\prod_{i=1}^{n}(X_{i})$ [closed]

P($X_{i}$=a)=P($X_{i}$=b)=0.5; a*b=0.5; $Y_{n}$=$\sum_{i=1}^{n}\prod_{i=1}^{n}(X_{i})$; Does {$Y_{n}$}$_{n=1}^{∞}$ converge a.s. to some limit random variables? If yes, what is its mean and ...
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24 views

What is the variance of $\frac{1}{n-1}\sum_{i=1}^{n-1}(x_{i+1}-x_i)^2$?

Consider a length $n$ vector $\mathbf{x}$ containing $n$ i.i.d. observations $\{x_i\}_{i=1}^n$ of a random variable $X$ with zero mean and unit variance. Let $\mathbf{z}$ be a length $n-1$ vector ...
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1answer
31 views

Finding the mean sales price

everyone. I have the following problem: a factory produce valves, with 20% chance of a given valve be broken. The valves are sold in boxes, containing ten valves in each box. If no broken valve is ...
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22 views

The following is my question. Thank you! [closed]

Suppose that a particle is released at the origin of the xy-plane and travels into the half-plane where x >0. Suppose that the particle travels in a straight line and that the angle between the ...
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1answer
41 views

Probability problem about repeated permutations

I have a box with $n$ red balls and $n$ blue balls, and I pick them one by one. When the color of the ball that I picked is different from the color of the previous ball, I earn a point. Let $X$ be ...
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17 views

Negative moments of variables

Let $f_i, i=1, \ldots, n$ be independent Steinhaus random variables, i.e. variables which are uniformly distributed on the complex unit circle. Let $a \in R^n$. Find $E\left(\sum_{i=1}^nf_i a_i\...
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1answer
17 views

Dependence of two random variables

In a Bernoulli experiment of parameter $p$ let $T$ be the instant of first success and $U$ the instant of second success. Find the density of $U$ and tell if $T$ and $U$ are independent or not. This ...
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1answer
19 views

a uniform random variable and its limit [closed]

I found this exercise on my book but I don't understand how to solve it: Let $U_n$ be an uniform random variable in $\{1,2,...,n\}$ and let $V_n=\frac {U_n}{n}$. Show that if, $0 \lt a \lt b \lt 1$ ...
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20 views

How to allow a genetic algorithm to build a generic mathematical formula?

Assume I have 4 known numbers, all in a 0-400 range, as follows: Variable1 Variable2 Variable3 Variable4 0-400 0-400 0-400 0-400 I also ...
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1answer
26 views

$X \sim U[-1,1]$, is there $Y$ independent of $X$ s.t $X+Y$ and $\frac{Y}{2}$ have the same distribution?

I have thought that if there exist such $Y$ then we can look at characteristic functions of $X + Y$ and $\frac{Y}{2}$ to get: $$ \phi_{X+Y}(t)= \phi_X(t)\phi_Y(t) = \phi_{\frac{Y}{2}}(t) \\ \frac{\...
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2answers
22 views

Almost sure convergence of random variables with same mean and the difference goes to zero on the product

Let $X_n$ be a sequence of independent real valued random variables on the same event space, with the same (finite) mean $\mu$. Suppose that for almost every couple of points $(\omega,\omega')$ in ...
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Understanding the Monte Carlo Equation which returns value to the expected function

I was reading an article related to Monte Carlo Method. The link of the article is: Monte Carlo Lecture I have following questions: 1)In the equation related to the attached image, we are assigning ...
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110 views

Understanding a common proof for linearity of expectation

For any two discrete random variables $X,Y:\Omega \to \mathbb{R}$ in a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, linearity of expectation tells us that: $$\mathbb{E}(X+Y) := \sum_{t \in (...
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Is there a meaningful additive risk measure

An important property of coherent risk measures is subadditivity. But are there any additive risk measures that can be used in a meaningful way? (I would exclude the expectation for example)
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Continuous variable function that evaluates to random curves

There exists a function of a continuous parameter function that sometimes evaluates to one curve and other to a different one? For instance, is there a function $f(x)$ that say is 0 for $x < 0$ and ...
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1answer
19 views

Two independent random variables X and Y are given. How can we compute Expectation minimum {X,Y} and Expectation maximum {X,Y}? [closed]

Two independent random variables X and Y are given. X is uniformly distributed in the interval [-2,5], Y is uniformly distributed in the interval [-4,2]. How can we compute Expectation minimum {X,Y} ...
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27 views

Let $m ≥ 1$, then I have to find $C_m > 0$ such that $f(n)$ = $\frac{C_m}{n}$ $−$ $\frac{C_m}{n+1}$ is a pmf on {${m,m+1,…}$}

I really could use some help with the following questions regarding Probability Theory. I really do not know where to start. Let $m ≥ 1$, then I have to find $C_m > 0$ such that $f(n)$ = $\frac{...
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33 views

Formalising some very basic probabilistic problem

Often, I encounter some problem were probability is involved. On an intuitive level, I understand I think what is going on but when I have to formalize it with probability spaces I am a bit lost. For ...
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1answer
29 views

Does the convergence in probability imply the following limit is $1?$

Let $X_n \in \mathbb{R}$ be a sequence of non-constant random variable with continuous PDF converging in probability to $c,$ but not necessarily convergence almost surely, i.e. $$\lim\limits_{n \to \...
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3answers
58 views

Distribution of $\Big(Y_1+Y_2\Big)^2$ and $\Big(Y_1-Y_2\Big)^2$ where $Y_i \sim N(0,1)$

Does anyone know what is the distribution of $(Y_1+Y_2)^2$ and $(Y_1-Y_2)^2$ where $Y_i \sim N(0,1)$ are independent variables? I have tried to go through the joint pdf, but when trying to change ...
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42 views
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An upper bound for $\sum_{n=1}^{\infty} n^{r-2} P\{|\sum_{k=1}^{n} \sum_{i=-\infty}^{\infty} a_{i} X_{i+k}|>\varepsilon n\}$

Let $1 \leq r<2$ and let $\left\{X, X_{i},-\infty<i<\infty\right\}$ be a sequence of pairwise i.i.d. random variables. Let $\left\{a_{i},-\infty<i<\infty\right\}$ be a sequence of real ...
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2answers
25 views

Brownian motion increments - are they random variables or random processes

If $W_t$ is a Brownian motion process and $0 \le t_1 \le t_2$ then is the increment $W_{t2} - W_{t1}$ a random variable or a random process? My lectures say "random variable" but I believe it makes ...
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1answer
29 views

Almost sure convergence of a sum of independent r.v

Let $S_n:=\sum_{i=1}^nX_i$ where $X_1,X_2,...$ are indepentent r.v.'s such that: $P(X_n=n^2-1)=\frac{1}{n^2}$ and $P(X_n=-1)=1-\frac{1}{n^2}$ Show that $\frac{S_n}{n}\rightarrow-1$ almost ...
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1answer
50 views

How to solve the below limit by means of L'H$\hat{\text{o}}$pital rule?

Let $(X_n)_{n\geq1}$ be a sequence of random variables satisfying: i) $\mathbb{E}\{X_n|\mathcal{F}_{n-1}\}=0$; ii) $\mathbb{E}\{X_n^2|\mathcal{F}_{n-1}\}=1$; iii) $\mathbb{E}\{|X_n|^3|\mathcal{F}_{n-1}...
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1answer
29 views

Inequality for the expectation of a nonnegative random variable

Let $X$ be a nonnegative random variable with distribution $F$ and mean $\mu=E(X)>0$. Let $A_{\mu}=[\mu, \infty)$. Is it true that $$ \int_{A_\mu} x dF(x) \geq \mu/2 $$ must then hold? I'm trying ...
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62 views

Probability to hit a set of pairs

Let $[n] = \{1, \ldots, n\}$, and suppose that $X = \{(i,j) : i< j\}$ is some subset of $[n]^2$, and we are given only its size $|X| = x$. Now assume I sample $y$ elements uniformly at random ...
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1answer
37 views

Is this identity on derivatives of random variables valid?

Is this identity valid? $a$, $b$, $c$ are all random variables. $$da = \int(\int d(a|b,c) db)dc$$
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19 views

Sigma-algebra generated by a sequence with zeros

Let $\{x_i\}_{i\in\mathbb{N}}$ be a sequence of zero mean random variables on a probability space $(\Omega, F, P)$. Suppose that there is some $i_0>0$ such that $x_i=0$ for all $i\geq i_0$. Does ...
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19 views

Show that : $\sum_{n=1}^{\infty} n^{-3} \sum_{k=1}^{n} E|X_{k}|^{2} I(|X_{k}| \leq n) \leq c \sum_{k=1}^{n} E|X_{k}|^{p} \sum_{n=k}^{\infty} n^{-1-p}$

I was trying to add some details to a result concerning the strong law of large numbers for pairwise independent random variables : here's what I've done : Let $\epsilon > 0,1<p<2, \, (X_i)_{...
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1answer
15 views

How do I obtain the pdf of a random variable, which is a function of random variable.

A random variable, $X$, has a value of zero with probability $1/3$, and follows a uniform distribution over $[-1, 1]$ with probability $2/3$. How can I derive the pdf of $X$? In my opinion, $X$ can ...
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1answer
36 views

Is it true that $\mathbb{E}\{X\}=\mathbb{E}\{Y\}$ $\Rightarrow$ $\mathbb{E}\{X|\mathcal{F}\}=\mathbb{E}\{Y|\mathcal{F}\}$?

Given $(\Omega$, $\mathcal{F}$, $\mathbb{P})$ and two r.v.'s $X$ and $Y$ defined on it, does it hold true that: $$\mathbb{E}\{X\}=\mathbb{E}\{Y\}\Rightarrow\mathbb{E}\{X|\mathcal{F}\}=\mathbb{E}\{Y|\...
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19 views

What is the CDF of a linear combination of a Gamma and a Hyperbolic random variables?

I am interested in the evaluation of the risk stemming from the combination of two weather events: heavy rain and low temperature. Rainfall $R$ can be modeled as a Gamma distribution, temperature $T$ ...
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2answers
24 views

Proof that two iid Gaussian random variables are conditionally independent of their mean given their sum

My question is a simplification of a statement in this book that i.i.d. Gaussian random variables $X_1, X_2, ..., X_n \sim \mathcal{N}(\Theta, 1)$ are conditionally independent of $\Theta$ given their ...
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3answers
37 views

Doubt about random variables and sample spaces

I have this little problem, reviewing my notes from the university of the first days of probability, I realized that there was a blank problem in my notes, says the following: " In a bag there are 5 ...
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3answers
50 views

If $X$ is a random variable, and $Y= 2X$, then why isn't it enough to multiply the density function of $X$ by $2$ to find the density function of $Y$?

This may be a dumb question, and I've tried searching online for answers, but I can't seem to wrap my head around it. So say I have a random variable $X$ and $Y = 2X$. Now I want to find the density ...
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1answer
23 views

No sequence of rv's such that $X_n\overset{P}{\rightarrow}0$ and $\mathbb{E}(X_n)\to 2$ and also $\sup\mathbb{E}(X_n^2)<\infty$.

Prove that there is no sequence $(X_n)$ such that $X_n\overset{P}{\rightarrow}0$ and $\mathbb{E}(X_n)\to 2$ and also $\sup_n\mathbb{E}(X_n^2)<\infty.$ Attempt. If we didn't have the last ...
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15 views

$\min(S^3,T)$ with $S \perp T$

Let $S$ and $T$ two random variables with exponential distribution of rate $\lambda$ and density $f(u)=\lambda e^{-\lambda u},u>0$. Find the density of: 1) $X=|S-T|$. $\rightarrow X\sim Exp(\...
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0answers
9 views

Stopping time for sum of draws from a uniform random variable [duplicate]

If you generate $u1,u2..$ from a uniform distribution in $[0,1]$. What is the average stopping time $T$ for the sum $(u1+..+uT)$ to be greater than $1$? Apparently the sum of random variables forms ...
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1answer
45 views

$\mathbb{P}(X>Y)$ for $X,Y$ two Poisson

Two teams have to play a final of a tournament. Team A score a number of goals that can be shaped like a random variable $X \sim Poi(\lambda_{A}=2.5)$. Team B score a number of goals that can be ...
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0answers
3 views

Independence relation for joint filtrations

Let $X$ be a random variable with first moment. Let $\mathcal A$ and $\mathcal B$ be sub-$\sigma$-algebras. Let $\mathcal A$ be independent of $\sigma(X) \vee \mathcal B$. Does it hold that? $$ E( X ...
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0answers
31 views

In what sense are independent random vectors “independent”? [duplicate]

Let's say we have independent random vectors $\boldsymbol{X}$ and $\boldsymbol{Y}$, where $\boldsymbol{X}=(X_{1},...,X_{p})$ and $\boldsymbol{Y}=(Y_{1},...,Y_{q})$. What is it that makes them ...
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1answer
39 views

Entropy of the Uniform Mixture of Discrete Probability Distribtuions

Consider the following inequality: \begin{equation} H\left(\frac{1}{3}p_{1} + \frac{1}{3}p_{2} + \frac{1}{3}p_{3}\right) \geq H(0.5p_{1} + 0.5p_{2}) \end{equation} where H(.) denotes the Shannon ...
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0answers
28 views

Expectation of maximum over an infinite sequence of sub-Guassian random variables

It's a problem from 'High Dimensional Statistics' (MIT's lecture notes) (Problem 1.3,page30)Let $X_1,X_2$ ... be an infinite sequence of sub-Guassian random variables with variance proxy $\sigma_{i}^...

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