# 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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### 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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### 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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### 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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### 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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### “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 ...