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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How to find marginal density from joint density

Let $$\begin{gather*} X \sim \text{Uniform}(-1, 1),\\ \varepsilon \sim \text{Uniform}(0, 1),\\ Y=X+\varepsilon X, \end{gather*}$$ where $\varepsilon$ and $X$ are independent. I need to find $f_{Y\mid ...
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What is the difference between a “population,” a “sample space,” an “underlying probability distribution? and a ”model"?

I'm trying to understand an overview of the topic of statistical inference. I have learnt bits and pieces of many of the probability and statistics involved in it but before learning it rigorously it ...
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Vocabulary clarification (Sample, Observation, Outcome, etc.)

As a math guy trying to understand Principal Component Analysis from the standpoint of Linear Algebra, I am following along with this paper I found. As I read it, I want to understand the statistics ...
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D+R: How many randomized trials to determine the D value?

I have an unknown fixed integer D. N binary random variables R were added to D. In each randomized trial we can observe the sum D+R. But we don't know D and we don't know the R's outcome in each ...
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Conditional probability involving black-box functions

Let X be a continuous random variable subject to a given probability density function. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be two black-box functions, ...
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Data regarded as a realisation of randomisers variables [duplicate]

In the following link, https://stat.ethz.ch/~geer/mathstat.pdf, author writes the following on pg.7. "The data consist of measurements (observations) $x_1, . . . , x_n$, which are regarded as ...
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How to develop a model in R to illustrate that sum of two variables normally distributed if these two variables each follow a Normal distribution? [duplicate]

I was wondering how can I use code in R to set up a simulation to show that if two random variables each follow a normal distribution with given means and variances, their sum is also normally ...
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Can the difference of random variables be uniform distributed? [duplicate]

Given two random variables X and Y with some distribution D, is it possible to choose a D such that Z = X - Y is uniform? Is there a standard distribution D that would satisfy this?
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Independence under linear transformations

What is the (largest) set of matrices $\mathcal C\subset \mathbb R^{m\times n}$ ($m\le n$) for which the following statement is true? Let $x_1,\ldots,x_n\in\mathbb R$ be independent random ...
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How to represent skewness(X) in terms of the expected value?

Let $X$ be the random variable. $E(X)$ is the expected value of $X$ Then $Var(X)$ = $E(X^2)$ − $[E(X)]^2$ where $Var(X)$ is the variance of $X$ Then how to represent skewness(X) in terms of the ...
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How to normalize one variable using another? [closed]

How do you normalize one variable using another? Note that this question is not about normalizing multiple variables but, rather, about basing the normalization of variable off the "weight" of another....
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How to find input to Gamma CDF which gives specific probability

I would like a formula which allows me to input some value for a Gamma distribution random variable, and get back the total probability density up to that point. In essence, I would like say, a ...
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How to interpret sum of two random variables that cross domains?

suppose we have two discrete random variables: $X: \{$6 sided dice rolls$\}$ $\rightarrow \{1..6\}$ (following uniform distribution) $Y: \{$coin flips$\}$ $\rightarrow \{0,1\}$ (following uniform ...
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Constructing random variable with specific expected value

Lets say I have a random variable $X$ and we don't know what the expected value is other than that it is some number $J∈[0,1]$. Lets say I'm interested in the expression $2J$. Without knowing $J$ I ...
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Conditional expectation of random variables defined off of each other

First of all, when we say that $X_n \sim \text{Unif}(0,X_{n-1})$, what does that mean, rigorously? Does it mean that for every $\omega \in \Omega$, $X_n(\omega)\sim \text{Unif}(0,X_{n-1}(\omega))$? ...
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Is there a continuous function that accepts a single uniform random variable and returns two independent uniform random variables?

I can define a function $f(X) = (Y_1,Y_2)$ that accepts a random variable $X$ with a uniform distribution on $[0,1]$, and returns two independent uniform random variables $Y_1,Y_2$. This function ...
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1answer
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Parameters are supposed to be distinct within a set of independent but not identically distributed random variables?

I'm confused here with one thing: If $(X_1,...,X_n)$ is not idenctically distributed, then doesn't this mean that the $\theta$'s are supposed to have subscripts $i$ in the pdf given above? I mean the $...
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1answer
26 views

sum of two proportions

I have 10 sets of 100 marbles. I have two processes that pick marbles. One of them usually picks between 2 and 15% of the marbles in the set. Another picks between 0 and 90% of the marbles in the ...
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1answer
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Why condition on either the r.v. $X$ or $Y$ and integrate over a product of pdfs rather a single pdf to find this probability density?

Let $X$ have the probability density $f_{X}(x)=\lambda e^{-\lambda x}, \;\; x>0$ and let $Y$ have the probability density $f_{Y}(y)=\lambda e^{-\lambda x},\;\; y>0.$ Find the probability ...
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1answer
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Is it possible to have dependent and exchangeable random variables? [duplicate]

Is it possible to have dependent and exchangeable random variables? Someone told me yes.
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Reciprocal of a binomial random variable

A production line produces faulty items independently at random with probability p. (a) Let X be the number of faulty items in a batch of 10. What is the distribution of X, and what is P(X = ...
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Expectation and variance notation for ratio of random variables [duplicate]

When we have a ratio of random variables, is their expectation/variance defined in the same way? That is, if we want to write out explicitly $E[\frac{X}{Y}]$ where X and Y are random variables, then ...
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1answer
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Sample data and its corresponding random variables

Simple question but I can't find the logic behind that. In many texts I see expressions like "Let $\{ x_1, \dots, x_n\}$ denote our sample data and $\{ X_1, \dots, X_n\}$ their corresponding random ...
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How to find the probability of a random variable being greater then another

So I have two independent random variables $X$ and $Y$, $Y$ ~ $U[0, 3]$ and the density function of $X$ is as follows: $f(x) = 1/3$ if $0\le x \le 1$ $f(x) = 2/3$ if $1\le x \le 2$ $f(x) = 0$ ...
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Why do stochastic processes involve time? [duplicate]

We define random variables as functions on a sample space $X(ω), ω ∈ Ω$. Here I do not see time being involved. A stochastic process is a family of random variables, but they also are functions of ...
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semi randomized order in k-fold cross validation with the R rminer package

I would like to conduct a k-fold cross-validation procedure (k=5) with the function "crossvaldata" from the R rminer package. However, I cannot divide the train and test sets into a completely random ...
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1answer
39 views

Generate jointly distributed random coefficients with given mean and variance from $N(0,1)$ and some matrix $L$

Suppose $\beta_1 \sim N(m_1,s_1^2)$, $\beta_2 \sim N(m_2,s_2^2)$ and $cov(\beta_1,\beta_2) = s_{12}$. Now generate draws of these random coefficients from draws of two independent standard normal ...
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1answer
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Find joint pdf table of two discrete independent random variables $X$ and $Y$

Given the pdfs of two discrete independent variables $X$ and $Y$, write the joint pdf. There is a property that $ if\ \ p_{XY}(x,y) = p_X(x)p_Y(y) \ \forall i,j \Rightarrow \text{X,Y are ...
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What does checking the “behavior of the likelihood on the boundary of the parameter space” mean

What does the latter mean? If someone could explain. I already showed in this question that $\hat \theta(X)$ is the MLE.
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Calculating the maximum likelihood estimator for the variance of iid normal RV's

I have that $X=(X_1,...,X_n)$ are iid RV's with $N$ $\sim (0, \theta)$ (Mean $0$ and variance $\theta \gt 0$. I would like to show that the Maximum Likelihood estimator for $\theta$ is given by $$ \...
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1answer
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Seeking an intuitive understanding of independence for random variables

I'm trying to gain a deeper, more intuitive understanding of what it means for two random variables $X$ and $Y$ to be independent. From my statistics courses, I know that $X$ and $Y$ are independent ...
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1answer
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example of when the likelihood function does not sum up, or integrate to $1$? [duplicate]

Could someone please give an example of when the likelihood function does not sum up, or integrate to $1$? I have seen this question with the first answer but it really confused me - why are we ...
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1answer
33 views

Non-independence in data and GLMMs

I am working with a dataset that consists of categorical variables and count data. My response variable is DPM SpeciesA (detection positive minutes of Species A, where for each hour I have a count of ...
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1answer
44 views

Fourth moment bound for unit-variance distribution

Given that a random real variable $X$ has zero mean and variance equal to 1, can we bound its fourth moment $\langle X^4\rangle$ (assuming it exists)?
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1answer
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GLM with random factors for observational design

I would love your help: I have 50 houses where I have counted the number of Mosquito's eggs once a week for 4 months. I have 5 fixed factors (temperature, NDVI, precipitations, etc) and I want to add ...
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1answer
50 views

Multiplying two markov chains

Let there be two homogenous markov-chains $(X_t)_{t \in \mathbb{N}_0}$ and $(Y_t)_{t \in \mathbb{N}_0}$ with transition matrices $P_X$ and $P_Y$, given as follows: $P_X = \begin{pmatrix} 0 & 1 &...
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Question about conventions for notation of random and regular variables in this regression example

I’m trying to follow some lectures notes on regression I found online and want to make sure I’m interpreting the notations correctly. On page 4 here, we have $\mu(x) = E[Y | X = x]$. Is it correct ...
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Stats Puzzle About Random Guesses [closed]

If you are told to guess a number between 1 and 100 that is as close as possible to the standard deviation of the guesses of others who are told the same thing, what number would you guess?
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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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1answer
26 views

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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1answer
47 views

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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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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1answer
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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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1answer
328 views

@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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1answer
41 views

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 ...
5
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1answer
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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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1answer
64 views

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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1answer
37 views

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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