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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 Probability distribution relates to neural networks?

The concept of random variables and probability distributions are confusing in the context of neural networks. In a neural network, which is the random variable and what is the probability ...
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47 views

Why data points are thought as random variable

I'am currently following a basic statistics course and a machine learning course. I try to understand what is covariance. In general, books define covariance as follows: Covariance is a measure of how ...
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22 views

Rewrite a constraint on the probability distribution using the cumulative distribution function

Consider a probability distribution $P: \mathbb{R}^3\rightarrow [0,1]$ and assume $$ (\diamond) \text{ }\int_{(x_1,x_2,x_3)\in \mathbb{R}^3 \text{ s.t. } x_3= x_1-x_2} dP=1 $$ Questions: Is there a ...
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45 views

How to generate random variables which are correlated and yet marginally identically distributed? [on hold]

I am wondering if there is a process to which we can generate random variables where there is a correlation structure between them, yet they are still marginally identically distributed? One idea that ...
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Visual basic code [on hold]

I want to produce 88 random numbers giving the sum equal to 440. With the following constraints. The numbers are between 0 to 20 (without 1) The numbers are integers I am searching for a VBA code ...
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31 views

Is there a distribution such that a sum of its samples creates the uniform distribution? [duplicate]

Say I sample $N$ numbers from a mystery discrete distribution and add them all together. My goal is for the sum of these random variables to be uniformly distributed from $[a,b]$ and thus the ...
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Why is a random variable being a deterministic function of another random variable mean that it is in the sigma algebra of the other variables?

Specifically, I'm learning about martingales in class right now. Given random variables $T$, $X_1, X_2, \ldots, X_n$, textbook that I'm reading draws an equivalence between the statement that the ...
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29 views

Theoretical question about randomness [closed]

Let's suppose we have to survey 30 males and 30 females in a town population of 100,000 by the phone. Every person in the town has one telephone number and we able to pick a number randomly. ...
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From trivariate cdf to the distribution of differences of random variables

Consider a trivariate cumulative distribution function (cdf) $G$. Is there a collection of necessary conditions on $G$ ensuring that $$ \exists \text{ a random vector $(X_1,X_2)$ such that $(X_1, ...
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1answer
85 views

For three variables $X,$ $Y,$ and $Z,$ prove that the sum of all the pairwise correlations is atleast $-3/2$

As I can understand about the question, it is required to prove, $$\operatorname{Corr}( X, Y ) + \operatorname{Corr}( X, Z ) + \operatorname{Corr}( Y, Z ) \geq -3/2 \tag i$$ But for any two ...
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Terminology for a “studentized” random variable?

Let $X_{1}, \dots, X_{n}$ be i.i.d. ramdom variables having mean $\mu$ and standard deviation $\sigma$. I wonder if the "studentized" $X_{i}$, the sample version of standardized $X_{i}$ where $\mu$ is ...
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80 views

Does independence between random variables imply independence between related events?

Say I have two random variables X1 and X2 and that they are independent. Am I guaranteed that the events "X1 is less than x1" and "X2 is less than x2" are independent? If not, under which conditions ...
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How to model the event occurring time?

Suppose I have some data (like following) to describe the time of certain event occurred. Which statistical model can be used to model this data? The goal would be predict when next event will happen. ...
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1answer
34 views

Doubt about state of Predictor Variable in PRF and its implications [Regression Question Series - Part 1]

Given a population $(X,Y)$ we hypothesize underlying population hasa regression line as follows. The conditional expectation is $$\begin{aligned} & E(Y|x) = \beta_0 + \beta_1x & \...
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1answer
41 views

probability that matrix $2\times2$ of Random variables is Invertible

Let $X_1, X_2, X_3, X_4$ to be Variables, and let $A$ be the following matrix: $$ \left[\begin{matrix} X_1 & X_2\\ X_3 & X_4 \end{matrix}\right] $$ assume that $X_1, X_2, X_3, X_4$ are ...
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1answer
39 views

Proving $X_n \rightarrow X$ in distribution implies $a+bX_n \rightarrow a+bX$ in distribution by definition

As stated in the title, I am trying to prove that if $X_n \Rightarrow X$ in distribution, then $a+bX_n \Rightarrow a+bX$ ( where $a,b\in\mathbb{R}$) in distribution using the definition as follows: $...
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1answer
32 views

Two dice roll with {1,2,3,4,5,6} and {10,20,30,40,50,60} and importance of RV mapping

We're all too familiar with a two-dice-roll experiment where we start with a uniform sample space of $S_{die}=\{1,2,3,4,5,6\}$ and end up in a non-uniform pmf for the sum of the numbers on the two ...
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Explain why $df(S_{xx}) = 2$ [duplicate]

Consider $X = \{x_1, x_2, x_3\}$. Then $\bar{x} = \frac{1}{3} (x_1 + x_2 + x_3)$ with degrees of freedom, $df(\bar{x}) = n = 3$. Now consider the total variation in $x$: $$S_{xx} = (x_1 - \bar{x})^2 ...
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1answer
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Should I use another random variable for the sum, if I work with random vectors?

Assume I have a simple random vector $(X, Y)$ with common distribution $P((0, 0))=1/6, P((1, 1))=1/6, P((3, 1))=1/4, P((0, 2))=1/6, P((1, 2))=1/4$, all others are zero. If I would like to argue ...
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38 views

Are relationships between distributions valid in both directions?

I know that if $X_1$ and $X_2$ follow a Gamma distribution with parameters $(p_1,a)$ and $(p_2,a)$ respectively, then $\frac{X_1}{X_2}$ will follow a Beta prime distribution of parameters $(p_1,p_2)$. ...
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If $X_n$ converges to $l$ almost surely and the $X_n$ are independent, must $X$ be a constant?

Let $T$ be a topological space and let $X_1,X_2,X_3 \dots $ be independent $T$-valued random variables. I want to know if the following implication is true: $$\mathbb{P}((\exists l):\lim_{n\to\infty}...
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1answer
28 views

How to find the minimum value of c? [closed]

In this book, the answer is not clear. What will be the method to find the value of c?
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1answer
55 views

Relationship between two Normal Random Variables with covariance matrices that are inverse of each other

Let $$x_1 \sim N(0, \Sigma_1)$$ $$x_2 \sim N(0, \Sigma_2)$$ $$\Sigma_1 = \Sigma_2^{-1}$$ That is each covariance matrix can be seen as the precision matrix (or inverse) of the other. Simple ...
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24 views

Random variable concept and terminology [duplicate]

I am a programmer with little mathematical background who started to study statistics/ML recently. I quickly stumbled upon the random variable term and it was hard for me to understand why in ...
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1answer
37 views

What is the distribution of $\frac{(Y_1 - Y_2)^2}{2},$ where $Y_i$ are standard Normal and independent.

Determine the distribution of $\frac{(Y_1 - Y_2)^2}{2},$ where $Y_i$ ~ $N(0,1),$ and $Y_1,Y_2$ are independent. I modelled the random variable in R and to me it seems like it's probably from a Gamma ...
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Long tailed count probability distributions - with available likelihood calculation and random number generation

in real world data, often (always to me) happen that for modelling count data Poisson Negative binomial multinomial dirichlet-multinomial probability distribution, are not robust to outliers, as ...
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1answer
37 views

Confusion with covariance

For distributions of random variables X and Y, their covariance can be defined as the difference between the multiplication of X and Y, normalized by their joint probability and the multiplication of ...
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1answer
37 views

P(X=x|Z=z) given Z=X+Y are all rv's

Let $X,Y,Z$ be random variables where $Z=X+Y$ and $X,Y$ are independent. By Bayes' Law, $$ \begin{align} P(X=x|Z=z) &= \dfrac{P(Z=z|X=x)\ P(X=x)}{P(Z=z)}\\ &= \dfrac{P(Y=z-x)\ P(X=x)}{P(Z=z)} ...
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Establishing an upper bound for the tail probability $P(X-\lambda \geq z)$ for any $z>0$, where $X$ is Poisson r.v. w/ parameter $\lambda$

Poisson random variable $X$ with the parameter $\lambda$ has, respectively, the pmf and the moment generating function of the forms $$P(X = k) = \dfrac{e^{-\lambda}\lambda^k}{k!}, \quad k=0,1,2,\...
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Marginal Distribution of Matrix Normal with Two Inverse Wisharts

Say I have a Matrix-Normal distribution and two Inverse Wishart Distributions $$X \sim MN_{p\times n}(0, \Sigma, \Omega)$$ $$ \Sigma \sim IW(a, A) $$ $$ \Omega \sim IW(b, B)$$ where $a$ and $b$ are ...
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1answer
24 views

How to perform a simulation of stock investment, capturing the variance?

I am simulating an individual, who invests throughout his lifetime in stocks and bonds. Bonds have fixed returns $r_f = 11\%$. Stocks are highly volatile and have returns $\mu = 22 \%$ and standard ...
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1answer
18 views

Find the mean, variance and others of a “RV = Gaussian RV + Discrete RV + ”?

Since I am solving a preparatory examen to study, it is not clear to me how to approach the question because I don't know what is the specific topic to study rigorously in order to know how to do it. ...
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24 views

How does the concept of consistency apply to the full bayesian posterior as opposed to a single estimate?

Towards the goal of making a bayesian statistical inference, I start by collecting $M$ independent and identically distributed data observations $D_i$. Then I take a Bayesian approach to learning the ...
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1answer
50 views

Upper bound on $P(n^{-1}\sum_{i=1}^n (X_i - \lambda_i)>t)$ for independent $X_i\sim\operatorname{Poisson}(\lambda_i)$

Let $X_1,\dots,X_n$ be independent random variables, $X_i \sim \operatorname{Poisson}(\lambda_i),$ $i=1,\dots,n.$ Let $$S=n^{-1}\sum_{i=1}^n X_i, \quad\quad \lambda=n^{-1}\sum_{i=1}^n \lambda_i.$$ ...
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27 views

How do I prove in this question that E(2^X) doesn't exist?

I understand how to get E(X), but how can I derive E(2^X)?
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1answer
28 views

Covariance between Linear Combinations of random vectors

Given a random vector $x\sim N(0, \Sigma)$ of dimension $p$ and matrices $A$ and $B$ (both $m\times p$, what is $Cov(Ax, Bx)$? It seems to me that the covariance should be $A\Sigma B^T$ but I am ...
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In general, how should we find the pmf given only the moment generating function without comparing its form to that of famous pmf?

Background It is known that moment generating function generates moments, but does it hold information about the probability of the random variable being realised at a particular value? Example ...
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1answer
38 views

Distribution of the dot product of a multivariate gaussian random variable and a fixed vector

If $a$ is a multivariate normal random variable, and $x$ is a plain old vector (of the same shape as $a$), then the inner product $x \cdot a$ is a random variable. This post on math exchange suggests ...
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29 views

Estimate unknown sum of iid random variables

Let $X_1, X_2, \dots$ be a sequence of independent and identically distributed discrete random variables with common mass function $f_X(x)$ defined for when $x \in \{0,1,\dots,N\}$ and $N$ is known. ...
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78 views

From bivariate to trivariate probability distribution

Let $\mathcal{G}$ be the space of all possible bivariate probability distributions. Let's pick a bivariate probability distribution $g\in \mathcal{G}$. Can we always find a random vector $(X,Y,Z)$ ...
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21 views

Necessary conditions for exchangeability

I have two claims that state some necessary conditions for exchangeability. I would like your help to understand whether they and relative proofs are correct. Consider three random variables $\...
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2answers
68 views

Why is expected value of random variable equal to mean

While learning about Random variables I came across the mean of random variable X. The definition says that the expected value of random variable E(X) = Mean of Random variable X I am not able to ...
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0answers
23 views

How to express the joint probability of two multivariate distributions over the same domain

I am working with multivariate normal distributions in scipy, but I have a question which is more statistics oriented: Say I have a multivariate distribution from two discrete random variables $X1$ ...
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0answers
44 views

Replacing summation by integral in classical variance of sum formula, is it possible?

It is well known that the variance of the sum of $x_1,...,x_N$ random variables is the sum of their variances plus twice their covariances: $\text{Var} \displaystyle\sum_{i=1}^{N}x_i =\displaystyle\...
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0answers
53 views

Expectation of the minimum of dependent random variables

How do we compute the expectation of the minimum of dependent random variables? In other words, what is the value of $\mathbb{E}[Y]$ in the following case: $$ \mathbb{E}[Y]= \mathbb{E}\big[\min(X_1,\ ...
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20 views

Random variable or sample space element?

The most common definition of a random variable goes something like "A random variable is a mapping $X:\Omega\to \mathbb{R}$ that assigns a real number $X(\omega)$ to each outcome $\omega$". So a ...
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2answers
70 views

Sample space for discrete random variables

Can discrete random variables be defined in a continuous sample space? Continuous sample space is a non countable sample space!
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20 views

Covariance of the inner product of a random vector and a constant vector?

Let $c$ be a constant $n$-dimensional vector. Let $Y$ be a $n$-dimensional random vector. What is: cov($c^\top Y$)? I do not understand this since surely $c^\top Y$ will return a $1 \times 1$ ...
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1answer
51 views

Dependence of estimator covariance on sample count

Say that $X$ is a set $\{X_1, X_2, ..., X_N\}$ of (non-independent) random variables, and that $\hat{\mu}$ is a set $\{\hat{\mu}_1, \hat{\mu}_2, ..., \hat{\mu}_N\}$ of estimators. Each $\hat{\mu}_i$ ...
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29 views

Determine the joint pdf of two new variables

I'm working on the following problem: and here's my attempt at a solution: From what I understand, $0<=x<=y<=1$ can be separated into $0<=x<=y$ and $x<=y<=1$, so when I plug in ...