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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In a multi-dimensional data array, which parts should be interpreted as the “random variables”?

Disclaimer: I am very much a beginner, so this question is probably confusingly written. Please bear with me. Background: I have a three-dimensional array of data, which was collected from a series ...
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Probability that a series of guesses from a normal distribution are non-random

Let's say I'm playing a game with my friends on the internet in which I guess their height. Furthermore, let's assume that the height in this population is normally distributed. At the end of this ...
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Calculating conditional probability $P(\Theta \le c | Y=0)$

Let $Y$ be a random variable with $Pois(\theta)$ distribution and the parameter $\theta$ be a realization of a random variable $\Theta$ with a priori distribution $Exp(\lambda)$. The task is to ...
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Find the covariance matrix of the random variables [duplicate]

What is the covariance matrix of $$f = 2x + 3y$$ if random variables x,y are independent and have a covariance matrices $\sum_{x}$ $\sum_{y}$? I know the covarince matrix of a random variable is ...
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Confirming meaning of Random Variable [closed]

Following several detailed posts on the topic of Random Variables What is meant by a “random variable”? Clarification - Random Variable Independent variable = Random variable? I would like to ask a ...
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Why Pearson correlation of two variables is very low but correlation between percentiles is almost perfect?

I have two variables say X and Y and when calculating Pearson correlation I found a very low value ~.01 but when I calculate the correlation between variables's percentiles, say Xp and Yp I found a ...
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Clarifying the meaning of i.i.d. when describing a set of variables

Let $Z_1, ..., Z_k$ be identically and independently distributed (i.i.d.) set of standard normal random variables. I understand that as part of the i.i.d. independent broadly means that variables ...
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Notation and meaning of a general probability distribution

I had two questions regarding the meaning and notation of a probability distribution when it is not specifically specified. For example, some papers jump right into their approach with notation like (...
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@whuber 's generation of a random variable with fixed covariance structure

The question refers to @whuber's algorithm to draw a random variable with a given covariance structure to a given set of random variables: https://stats.stackexchange.com/a/313138/3277 The algorithm ...
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2answers
36 views

Use the law of iterated expectations to prove the law of total covariance: [closed]

How to prove the law of total covariance using the law of iterated expectations? \begin{align} \text{Cov}(X,Y)=\mathbb{E}\big[\text{Cov}(X,Y\lvert Z)\big]+\text{Cov}\big[\mathbb{E}(X\lvert Z),\mathbb{...
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Binomial & numerical variables as dependent and independent + random variable

I am new to statistics and trying to figure out how to analyze my data correctly. I completed a biological study with the following variables: (I have converted my binary variables to 1 and 0) -Type ...
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$cor(B_1,Y) > cor(B_2,Y) > 0$ but $cor(A + B_1, A+Y) < cor(A + B_2, A+Y)$. Is this possible?

When I was processing data I came across this strange phenomenon. Say I have time series with positive values only, $A, B_1, B_2, Y.$ $\operatorname{cor}(X,Y)$ is the correlation of $X,Y.$ Here I ...
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Inverse transform sampling and ambiguous Intervals

Let $F_i:\mathbb R\to[0,1]$ be a distribution function$^1$ and $$F_i^{-1}(t):=\inf\left\{x\in\mathbb R:F_i(x)\ge t\right\}\;\;\;\text{for }t\in[0,1].$$ I've got a computer program where only $F_i^{-...
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I have two sampling techniques $\varphi_1,\varphi_2$. Given $x=\varphi_1(u)$ can I compute a $v$ with $x=\varphi_2(v)$?

I have two sampling surjective techniques $\varphi_1,\varphi_2:[0,1)\to E$ mapping a random number $u\in[0,1)$ to a sample in a measurable space $(E,\mathcal E)$. Say $u\in[0,1)$ and $x:=\varphi_1(u)$...
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What is the relationship between the distribution fitted to random samples of two different distributions? [closed]

Suppose I got two distributions (gaussian, for instance) $G_A$ and $G_B$. Now, I draw $n$ random samples from $G_A$ and $n$ random samples from $G_B$. Now I fit a distribution $G_C$, from the same ...
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Checking Condition for Independence of Random Variables [closed]

Which of the following conditions imply independence of the random variables X and Y? A. P(X >a I Y >a) = P (X> a) for all a in R. B. P(X >a I Y < b) = P (X > a) for all a, b in R. C. X and ...
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Predict vector of random variables from historical data?

There's historical prices for gold, sp500, silver, iron. ...
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Sum of two Von Mises random variables

I know that the sum of two independent normally distributed random variables is also a normal random variable, but is this true of other distributions? For example, what probability distribution does ...
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Translating model accuracy to probabilistic language

I have inputs $x$ which are transformed by some (parametrized ) function $g$ to give $h=g(x)$. I also have binary labels $z$. I know the following holds: for any model $f$, the expected accuracy of $...
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How can we write the below characteristic function?

Let us assume that $X$ is a random variable and $a$ is a constant. Now suppose $Y=a+bX$, what would the characteristic function of $Y$ would be? Is it? \begin{eqnarray} \mathbb{E}_X\left[\exp(iuY)\...
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Reference Request - Bounded Discrete Multivariate Stochastic Processes

I would like to be referred to papers or textbooks about the dynamics of non-negative discrete stochastic processes under the constraint that all variables sum up to some constant. Appropriate ...
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36 views

Expected value of function of dot product

Given a random vector $\bar{a}$ with uncorrelated random components of unit variance and zero mean, how do I calculate the expected value over the distribution $$\langle g(\bar{a}\cdot\bar{b})\rangle_{...
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1answer
47 views

Expected value of a conditional Y given X, $E(Y|X)$ is or is not a constant?

For a random variable $X$, there is an expected value $E(x)$. Since $E(X) = \mu \in \mathcal{R}$ where $\mu$ is a mean, and can be viewed as a constant. If this is true, then $E(E(X)) = E(\mu) = \mu$ ...
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What does the minimum of a random variable mean?

Let $X_1, X_2, X_3, \cdots,X_n$ be independent and identically distributred (iid) random variables. Then, how would you know/calculate what $min(X_1, X_2, X_3, \cdots,X_n)$ is?
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Differences between realization of the random variable and deterministic variable?

The first question is that can we classify variable into random variable and deterministic variable? The second question is that The possible values taken by a random variable"X"(Uppercase) are termed ...
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Why is the probability mass function of a transformed discrete random variable summed over the inverse values of the function?

Let $X$ be a random discrete variable with probability mass function (pmf) of $p_X(x) = P(X = x)$. Let $Y = g(X)$ (from $\mathbb{R}$ to $\mathbb{R}$). Then, why is it that: $$p_Y(y) = \sum_{x \in g^{-...
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1answer
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Hierarchical model for A/B experiment?

I'm new to Bayesian statistics. I have a metric that has a very non-parametric distribution, which would make it very difficult to use in an A/B experiment. However, it can be broken up into ...
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Intuition for seperable process

I am learning about seperable processes as defined in the below link: https://www.encyclopediaofmath.org/index.php/Separable_process Whilst I understand the maths definition and can see when a ...
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1answer
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How to relate the distributions of these trigonometric functions of uniform variables?

$\theta_1,\theta_2$ are two independent random variables distributed uniformly in $[0,2π)$. Let $X=\cos\theta_1,Y=\cos\theta_2.$ Prove that $\frac{X+Y}{2}$ and $XY$ are equal in distribution,$\frac{X+...
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Generate Multivariate Log-Normal Variables with given Covariance and Mean

Let ${\bf X}=(X_1,...,X_n)$ be an $n$-dimensional log-normal random variable. I want to $force$ my random variables to be such that $Cov(X_i,X_j)=\Sigma_{i,j}$ and $E(X_i)=\mu_i$ where $\Sigma_{i,j}$ ...
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Conditional expected value variable change

Assume that we have $X_1,X_2,X_3$ that are jointly Gaussian and with non-zero covariances. We also have: $$Y_1=X_1$$ $$Y_2=X_2−X_1$$ $$Y_3=X_3−X_2$$ With $0$ covariances among all $Y_i$ and $E(Y_i^...
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1answer
26 views

Moment of a function of Gaussian random variables: $\mathbb{E}[(a_{i}^{\top}AA^{\top}a_{j})^{q}]$

Let $A$ be an $m\times k$ matrix with iid $\mathcal{N}(0,1)$ entries and $a_{i}$ and $a_{j}$ be its $i$th and $j$th columns. I would like to compute the following quantity: \begin{equation} \mathbb{E}...
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What does it mean that random variables are “drawn from the same distribution”?

In the second bullet point, what does it mean that "$X_1,X_2,...X_n$ are drawn from a common distribution"? Does it simply mean they all have the same type of distribution (e.g. they are all normally ...
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Infinite discounted sum of betas

Let $0 \leq \gamma < 1$, $X_i \sim \text{Beta}(\alpha, \beta)$, and $$Y \sim \sum_{i = 0}^\infty \gamma^i X_i$$ What is the distribution of $Y$? Does it have a closed form? Can it be sampled ...
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Question about random variables and the distribution of the sample mean

I'm new to statistics. I am so confused as to why the Xbar (the random variable describing the sample mean) can be found by taking the average of all the X's. From what I understand the capital X's ...
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How generate random data that satisfy specific constraints such as having specific median? in R [closed]

In general, how can I simulate data that exactly satisfies a set of constraints? I'm in need of generating a set of random numbers, which conform to a given median (not mean), and also fall within ...
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How do you prove that samples are equally distributed even when taken without replacement? [closed]

I was reading Casellar and Berger's book and came across this well-known property. Although, in the book, the proof for any Xi is not provided. Does anyone know how to follow up on it? I'm curious, is ...
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Random variable vs Statistic? [duplicate]

What's the difference between a random variable and a statistic? It seems that formally, a random variable is simply any real-valued function (and its domain is a set that we call a "sample space"). ...
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Does covariance of $X$ and $X^2$ depend on the range of $X$?

Consider the random variables $X$. First suppose that $X\sim U(0, 1)$ (i.e. it has uniform distribution over $[0, 1]$). By simple transformation, I found that the density for $Y = X^2$ is: $p_Y(y) = \...
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Independence of random variables and sums of random variables

I am seeking to find the joint distribution of X and Y. I have the marginal distributions of X and X+Y and they are independent. We have that $f(X=x,Y=y)=f(X=x,X+Y=x+y)$ which is equal to $f(X=x)f(X+...
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Does the joint pdf $f_{x, y} (x, y)$ equal to the conditional $p_{y | x} (y | x)$ for all random variables?

So I have this question where you are given two random variables, $X$ and $Y$. $X$ is a continuous random variable (represented as a mean) with a distribution of $Exp(1)$ (exponential with $\lambda = ...
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1answer
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What is the purpose of doing Arithmetic(+-*/) on two different distribution?

I'm now learning mathematical-statistics, and I learned a lot of example, like " $X $and $Y$ are two independent gamma distribution, please prove that the addition of two gamma distribution is still ...
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Are all ergodic random processes (at least wide sense) stationary?

If not, please provide a simple example of a non-stationary process that is ergodic (in mean and covariance).
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How can I calculate Nagelkerke pseudo r2 with glmmkin object from GMMAT package in R?

I have used glmmkin function from GMMAT package to fit a logistic mixed model with the binary phenotype 'disease', one fixed variable 'PRS' and one kinship matrix 'GRM' to model the covariance ...
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Expected wait time in two different types of banks?

Problem Statement There are two banks both have 100 people inside who want to cash their check and must do so by talking to the teller. In both banks, there are 10 tellers. In Bank 1 all 100 people ...
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3answers
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How to generate a vector that has zero correlation with another vector (in R)?

Suppose I have a vector v1 with values in the set {-1,0,1} ...
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81 views

Finite $k$th moment of a function of random variable

Let $X = a/h$, where $X$, $a$ and $h$ are random variables, with $X$ an i.i.d. sequence. If $X$ has finite 8th moment, can we infer that $a$ has finite $8$th moment as well? Thanks
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“Stable” distributions with integer support?

My understanding of stable distributions is that in order for a distribution to be stable, linear combinations of independent random variables from a given distribution (for example, a Gaussian) must ...
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1answer
37 views

What are the conclusions to be drawn when a t-test is significant but a linear mixed effects model is not?

I have 30 participants. They have a pre score and a post score. I am testing whether this changes. There are five observations per participant. When the data are analyzed using a t-test there is a ...
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1answer
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Decomposition of auto correlated variable

Can the sum of two (or more) random variables with zero auto-correlations and zero cross-correlations yield a random variable that has non zero auto-correlation?