Questions tagged [random-variable]

A random variable or stochastic variable is a value that is subject to chance variation (i.e., randomness in a mathematical sense).

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119 views

How to generate two correlated random samples, one follows geometric Brownian motion, the other follows a beta distribution? [closed]

I'd like to conduct a Monte Carlo simulation with two random variables. One random variable is generated by geometric Brownian motion, the other random variable is sampled by drawing random values ...
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42 views

Laplace-Stieltjes Transforms and distribution

I was going through a paper, I came across below relation, \begin{equation} T=\begin{cases} C, & \text{with probability $P(H<C)$}\\ 0, & \text{with probability $P(H>C)$} \...
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35 views

Covariance of X and Y conditional on X+Y>Z? [closed]

Suppose that $X$, $Y$, and $Z$ are three independent random variables. Is there a way to compute the following conditional covariance? $Cov(X, Y | X + Y \geq Z)$
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Given $Z\perp X\mid Y$, is it true in general $Var(Z|h(X,Y))=Var(Z|h(c,Y))?$

Given random variables $X, Y, Z$: If $Z\perp X\mid Y$, then I know that $Var(Z|X,Y)=Var(Z|Y)$ But is it still true in general that $$Var(Z|h(X,Y))=Var(Z|h(c,Y))?$$ here $h$ is a real valued function ...
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26 views

Does entropy have less estimation error than mean and variance estimates?

Estimating the mean or expected value of a continuous random variable's (r.v.) empirical distribution is known to be difficult, moreso than estimating the variance. Estimates of the mean and variance ...
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60 views

What does maximizing mutual information do?

In information theory, there is something called the maximum entropy principle. Are other information measures, such as mutual information, also commonly maximized? If mutual information describes the ...
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56 views

Is a bounded real-number random variable discrete or continuous?

A discrete random variable is countable (such as integers and natural numbers), whereas a continuous r.v. is not countable (like the real numbers $\mathbb{R}$). If I have a dataset whose observations ...
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Is it possible to test for strongest predictor, if all variables are identical within a random effect grouping?

I'm trying to understand the regression analysis done in a published study. They use GAM models to predict plant growth, and to test for the most important predictors. They have measured growth of ...
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8 views

Estimating significance of a betting strategy, given probabilities of each event

We are given a set of independent Bernoulli trials, each having separate claimed probability (which might not be accurate). We are also given a single observation for each of those events. How to ...
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35 views

Expected value of indicator variables

I have an ensemble of graphs with $N$ nodes and $M$ undirected edges each. Let $X_{ij}$ be indicator variables that point to the existence of an edge between nodes $i$ and $j$ on a particular graph. ...
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43 views

$\inf$ of a sequcence of random variables bigger than some $a\in\mathbb{R}$

Suppose we have sequence of random variables $\{X_n\mid n\in\mathbb{N}\}$, defined on a probablity space $(\Omega,\mathcal{F},\mathbb{P})$. Then we define $(\inf_{n\in\mathbb{N}}X_n)(\omega)=\inf_{n\...
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Controlling the influence of variation in a factor

I sampled communities in different habitats along a continental margin. Depth is a strong structuring factor in the ocean. To mitigate or control the influence of the depth factor and in a comparison ...
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48 views

Central limit theorem - num random variables vs. sample size?

Does the Central Limit Theorem require the number of random variables to increase to a sufficiently large number or the number of samples of each random variable to increase to a sufficiently large ...
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28 views

Probability density function after transformation

Let $X,Z$ be random variables with probability density functions $p_X,p_Z$. Suppose $Z=f(X)$, where $f$ is continuous and differentiable. How is $p_Z$ related to $p_X$? It's tempting to say $p_Z(z) ...
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464 views

Sigma algebra generated by random variable on a set with generators

I'm having trouble proving an intuitive result I found in these lecture notes I'm using for self-study (1.2.14 there). Suppose $X$ is a $(\mathbb{S}, \mathcal{S})$-valued random variable (from $(\...
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Which method to use to test the effect of a dichotomous independent variable on a continuous dependent variable?

My apologies in advance if there are fundamental mistakes or wrong assumptions in my question. Also I am making up the example so there may be inconsistencies. I’ll explain this with an example. I ...
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Find an expression for the cumulative distribution function of a random variable

Let $X$ have distribution $F$ and density function $f$ and let $A$ be a subset of the real line. Let $I_A(x)$ be the indicator function for $A$: $$I_A(x) = \begin{cases} 1 \hspace{5mm} x \in A\\ ...
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Are two random vectors independent if their corresponding components are all independent?

Let $\mathbf{X} = (X_1,\ldots,X_n)$ and $\mathbf{Y} = (Y_1,\ldots,Y_n)$ be random vectors, and let $f_{\mathbf{X}}(x_1,\ldots,x_n)$ and $f_{\mathbf{Y}}(y_1,\ldots,y_n)$ be their respective pdfs or ...
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find parameter a so that X+Y and X+aY are uncorrelated

Given random variable X and Y such that E(X)=0, E(Y)=-1 Var(X)=1, Var(Y)=4 Var(X+Y)=9. Find parameter a so that X+Y and X+aY are uncorrelated. The answer is -1/2. Would appreciate any tips on how to ...
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Compute 16 var(x)+32 var(y) for given bivariate CDF

\begin{equation} {F(x,y)} = \begin{cases} 0 & \text{if $x<0$ or $y<0 $} \\ \frac{1-e^{-x}}{4} & \text{if $x>0, 0 \leq y <1$} \\ 1-e^{-x}& \text{if $x \geq 0, y \...
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Doubt in independence of 2 random variables

If 2 random variables are independent, then $f(x,y) = f(x)f(y)$. Is converse true? $F(x,y)=F(x)F(y)$. Is converse true? $E(x,y)=E(x)E(y)$. Is converse true? where $F$ is cdf and $f$ is pdf I recently ...
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cosine of angle between random variables is equal to the correlation coefficient? [duplicate]

I have seen this said multiple times where (1) the cosine of the angle between the random variables (on a vector space) is equal to the correlation coefficient, and (2) the claim if random variables ...
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What does it mean if X → Y → Z is a markov chain, it mplies that Z → Y → X. Some-times written X ↔ Y ↔ Z

In the book Elements of information theory (Cover, Thomas) 2nd ed. Page 34 On Markov Chain it says: ...
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How to think of real life data in terms of random variables?

If I have a variable in my linear model (for example, age of person). Let's say my sample is {25,30,50,54,26,19}. Would this sample be the sample from a single random variable or is this a sample of ...
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Should “City” be a fixed or a random effect variable?

I am analyzing data on "BloodSugar" level (dependent variable) and trying to find its relation with "age", "gender" and "weight" (independent variables) of ...
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11 views

Are populations related to random variables when discussing parameters?

The term "parameter" (as opposed to a statistic) is defined as a value used to describe a population, but it's also defined as a value used to describe the distribution of a random variable. ...
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15 views

How to generate 3 correlated standard normal random variables given pairwise correlation?

I know how to generate normal random numbers given a covariance matrix. However, I am trying to generate sets of correlated standard normal random numbers given pairwise correlations. It is easy when ...
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What does “A realization of independent copies”? [duplicate]

I read in statistic, "a realization of independent copies" from "Elements of copula modelling with R", and do not understand the meaning. I search and found that it means, the two ...
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172 views

Law of unconscious statistician for conditional expectation

I have random variables $X$, $Y$ with joint distribution $f_{XY}(x,y)$ and conditional distribution $f_{X|Y}(x|y)$ and another random variable $Z=g(X)$ with $g$ being bijective is it true that $$E(Z|Y=...
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Prove a result on expectation with 2 random variables

Part 1 I wish to show that: $$E(X)=E[(X|Y=y1)*Pr(Y=y1)+(X|Y=y2)*Pr(Y=y2)]$$ where the random variable Y can take 2 possible values. As many comments and answers have suggested, this may be technically ...
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Is a sample i.i.d or is a collection of random variables i.i.d.?

Basic terminology question. I hear “let the sample be i.i.d.“ and “let these random variables be i.i.d.” being used interchangeably. Even Wikipedia uses both: A collection of random variables is ...
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31 views

Is it possible to to have more than 2 groups for biased coin randomization? If no, is there any modifications supporting multiple groups?

Is it possible to have more than 2 groups for biased coin randomization? If no, is there any modifications of biased coin randomization supporting multiple groups? This is my R code for generating ...
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28 views

How to compute $\mathbb{E}[\|\frac{1}{K}\sum_{k\in\cal{K}}\bf{X}_k-\bf{X}\|^2]$?

Assume vector $\bf{X}_k\in\mathbb{R}^N$, for all $k\in\cal{K}=\{1,2,\cdots,K\}$ are $K$ i.i.d. random variables. If each $\bf{X}_k$ is an unbiased estimator of parameter $\bf X$, then how to compute ...
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34 views

Definition indepedence and identically distributed (iid)

Bruce Hansen's book "Econometrics" defines a random sample as follows: "The observations $(y_i,x_i, z_i)$ are a random sample if they are mutually independent and identically ...
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Conditional Gaussian from Joint Distribution

I have the random variables $X$ and $Y$ related through the following: \begin{align} X &= N_X \\ Y &= 4X + N_Y \end{align} where $N_X, N_Y \overset{\text{iid}}{\sim} \mathcal N(0,1)$. Is ...
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Convergence in Probability (Analytical Solution Verification)

Problem: Let $X_1,X_2,\cdots$ be independent random variables that are uniformly distributed over $[-1,1]$. Show that the sequence $Y_1,Y_2,\cdots$ converges in probability to some limit, and identify ...
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Distribution of sum of Indicator Variables

Let $X1 $, $X2 $, $X3 $...$Xn $ be n observations with distribution function $F $. Let $F^{*} $ be the empirical distribution of the random sample. $F^{*} = \frac{1}{n} \sum I(X_{i} \le x)$ where I = ...
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Does the mean preserving spread of a distribution constitute a mean preserving spread of the joint distribution of two iid draws from it?

Let random vectors $X_1, X_2 \sim F, \;i.i.d, X_1, X_2 \in X $. Now replace $F$ with its mean-preserving spread (MPS), say $G$. My question is, does that constitute an MPS of the joint distribution of ...
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Convergence example

I'm studying convergence in my probability class and I'm asked to show if there exists any convergence for the following sequence of random variables: $$\left\{\frac{W_n}{ln(n)}\right\}_{n\geq1} \ s.t....
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How to estimate parameters and pdf of a random variable transformed from a lognormal random variable?

I have a continuous random variable Y that follows lognormal distribution with known parameters (mu and sigma). Let Y be transformed to X=Y-20000. So it is basically shifted to left. How do I find the ...
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How is the relation of expectations and order statistics of X and Y, if Y is a mean-perserving spread of X

I am searching now for quite some time for an answer with a source to the following question. $X$ and $Y$ are two random variables. $Y$ is a mean preserving spread of $X$. Now, how is the relation of ...
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Simulate correlate random variables with given marginal distribution where one is always larger

Is it possible to simulate pairs of random variables with a given marginal distribution and population correlation where one random variable is larger than the other? More formally, I need to simulate ...
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106 views

Distribution of X+U when X is a discrete and U is a continous random variable

Suppose $X$ and $U$ are independent random variables. $X$ is a discrete uniform variable and $U$ is a continuous uniform $[0,1]$ variable. What is the value of $\mathbb P(X+U\leq y)$, where $y$ is a ...
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Any transformation to make a process covariance stationary

In a typical AR model, for simplicity take AR(1), we know that if it is of the form $x_{t}=\alpha_{0}+\alpha_{1}x_{t-1}+\epsilon_{t}$, it may not be mean stationary as $\alpha_{0}$ is 'reinforced' by ...
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Expectation of Mixed Random Variable (Contradiction with Manual Solution)

$X \sim \mathcal{N}(1,\text{negligible variance})$ and $Y \sim \mathcal{N}(2,\text{negligible variance})$ \begin{equation*} Z= \begin{cases} X, & \ \text{w/pr}\quad p\\ Y, &...
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1answer
45 views

A linear process $x_{t}$ satisfies $\sum\limits_{j \in \mathbb Z}\lvert \gamma(j) \rvert < \infty$

A linear process $x_{t}$ is the weighted sum of white noise variates $(w_{t})_{t}$, i.e. $$x_{t}=\mu+\sum\limits_{k \in \mathbb Z}\psi_{k}w_{t-k}$$ such that $$ \sum\limits_{j \in \mathbb Z}\lvert \...
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Question about Functions of Several Random Variables

In the Mathematical Statistics and Data Analysis by John Rice, it states that for random variables $U,V$ which are functions of random variables $X,Y$, we have: We know that $$f_{UV}(u,v) = f_{XY}(h_1(...
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101 views

Random variables $(X,Y)$ with $\text{Var}(X)<\text{Var}(Y)$ and $\mathbb{E}(|X-\mu_X|)>\mathbb{E}(|Y-\mu_Y|)$

I am looking for an example of a pair of random variables $(X,Y)$ with expected values $(\mu_X,\mu_Y)$ satisfying the following relationships: $$ \text{Var}(X)<\text{Var}(Y) $$ and $$ \mathbb{E}(|X-...
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1answer
41 views

Difference between random normal variable & a random number

What is the difference between drawing 10 random normal variables & randomly picking 10 numbers For e.g. if I generate 10 random normal variables & the result is ...
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40 views

Questions about smoothness and determinacy of stochastic processes

Recently I've been dealing with a problem involving stochastic processes. However, I found myself not so familiar with this topic. I have the following two questions regarding whether there is a ...

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