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).

Filter by
Sorted by
Tagged with
0
votes
0answers
14 views

Predict vector of random variables from historical data?

There's historical prices for gold, sp500, silver, iron. ...
0
votes
1answer
57 views

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 ...
0
votes
0answers
18 views

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 $...
1
vote
0answers
21 views

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)\...
0
votes
0answers
19 views

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 ...
1
vote
0answers
38 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_{...
1
vote
1answer
51 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 $E(E(X)) = E(\mu) = \mu$ ...
3
votes
3answers
92 views

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?
0
votes
1answer
30 views

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 ...
2
votes
2answers
48 views

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^{-...
2
votes
1answer
42 views

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 ...
0
votes
0answers
5 views

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 ...
2
votes
1answer
43 views

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+...
1
vote
0answers
17 views

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 $\Sigma_{i,j}$ ...
1
vote
0answers
29 views

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^...
1
vote
1answer
28 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}...
0
votes
0answers
30 views

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 ...
4
votes
1answer
98 views

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 ...
3
votes
3answers
52 views

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 ...
1
vote
2answers
27 views

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 ...
1
vote
0answers
14 views

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 ...
9
votes
3answers
511 views

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"). ...
0
votes
2answers
47 views

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) = \...
1
vote
0answers
34 views

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+...
1
vote
2answers
38 views

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 = ...
1
vote
1answer
28 views

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 ...
1
vote
1answer
39 views

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).
0
votes
0answers
56 views

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 ...
0
votes
0answers
23 views

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 ...
1
vote
3answers
68 views

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} ...
3
votes
2answers
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
1
vote
0answers
15 views

“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 ...
0
votes
1answer
38 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 ...
2
votes
1answer
25 views

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?
0
votes
1answer
28 views

Random Walk Stopping Time Calculations

Let $S_n$ be a random walk with $P(S_{n+1}=S_n+1|S_n)=p<\frac{1}{2}$ and $1-p=q=P(S_{n+1}=S_n-1|S_n)$. Let $\tau=min(n:S_n=0)$ How may we show that for any positive integer $x,\mathbb{E}[\tau|...
3
votes
1answer
44 views

How to generate a conditional subset?

I want to develop an algorithm to generate a random subset (size $k$) from $\{1, . . . , n\} $ given that it contains at least one of the elements in $\{1, . . . , s\}$ ($s,k\ll n$). This is what I ...
0
votes
2answers
35 views

Is there a code/ software that generates random numbers? My specific interest - from Dirichlet distribution [closed]

Is there a code/ software that generates random numbers? My specific interest - from Dirichlet distribution. Lets say, x1,x2,...x8 and 1000 random instances of it with a uniform alpha=2.
0
votes
0answers
14 views

Range of correlation coefficient [duplicate]

Suppose there are $n$ random variables $X_1, X_2, \ldots, X_n$ If the correlation coefficient of $X_i$ and $X_j$ equals a constant $\rho$ for all possible choices of $i,j$ such that $i \neq j$ then ...
1
vote
1answer
43 views

Proving that as N increases, it is more likely that x - y > z, with x \in X, y \in Y, z \in Z

In trying to solve a bigger applied problem, I found myself facing the following. Let $X$, $Y$ and $Z$ be three independent random variables, each coming from an unknown distribution, and each with $...
0
votes
0answers
19 views

Variance of estimator of stratified sampling

Consider a stratified design composed of $H$ strata of size $N_h$, $h = 1, . . . ,H$. We want to estimate the population mean $μ_y$ of the characteristic $y$. Let $μ_{x,h}$ $h = 1, . . . ,H$ be ...
-1
votes
1answer
40 views

I want to represent x1, x2, …, xn (where their sum =1) by Dirichlet distribution. What alpha's should I select if x1, x2,… have the same pdf

I want to represent x1, x2, ..., xn (where their sum =1) by Dirichlet distribution. What alpha's should I select if x1, x2,...,xn have the same probability density function? all 0 < xi < 1. In ...
1
vote
0answers
35 views

Show that if $Y$ has $E(Y)=μ,Var(Y)=σ^2$, and $P(|Y-μ|<σ)=0$, then $Y$ has the same distribution as $X$ described below for $k=1$

I'm having a hard time proving the second and third case of $X$: Show that if $Y$ has $E(Y)=μ,Var(Y)=σ^2$, and $P(|Y-μ|<σ)=0$, then $Y$ has the same distribution as $X$ described below for $k=1$. $...
0
votes
0answers
39 views

Calculation of joint PDF

we have the joint PDF of two RVs $X$ and $Y.$ we also have two RVs $U = f(X,Y)$ and $V = g(X,Y),$ where $f$ and $g$ are two variable functions. How can I calculate the joint PDF of $U$ and $V$? for ...
3
votes
3answers
2k views

What is the “opposite” of a random variable?

I am learning about random variables with all of their different types of distributions for discrete and continuous types. However, before knowing about random variables, I am not sure what would be a ...
1
vote
0answers
13 views

Reference request: statement about jointly Gaussian RV

I found a theorem online https://people.eecs.berkeley.edu/~ananth/223Spr07/jointlygaussian.pdf Is there a secondary reference for this theorem?
0
votes
1answer
20 views

Order dependencies between binary variables

I have a series of random, binary variables that are sampled in a particular order. Some may be affected by the order in which they're sampled, i.e. there may be order dependencies. Suppose I sample ...
3
votes
1answer
15 views

Is there an index that quantifies the degree of independence of dependent random variables?

Is there an index that quantifies the degree of independence of dependent random variables? Suppose there are n dependent and identically distributed variables, I was thinking of the ratio covariance/...
2
votes
0answers
21 views

Generating Random Variables from the Generalized Beta Distribution

There is a very nice solution to generating random numbers from the Generalized Distribution of the second kind which can be found here. There is a more general form of this which was developed by ...
0
votes
0answers
33 views

Independence of a Gaussian random variable and the product of another Gaussian random variable and a Bernoulli random variable

Let $X$ and $Y$ be two independent Gaussian random variables with mean $0$ and variance $σ^2_X$ and $σ^2_Y$ respectively. Let $Z$ be a random variable measurable with respect to $σ(Y)$ and suppose ...
0
votes
2answers
40 views

Expectation of inner product between random vector and constant vector $\mathbb{E}_{{\bf{\epsilon}}}[\langle {\bf{x}}, {\bf{\epsilon}}\rangle] = ?$

Suppose I have a random vector ${\bf{\epsilon}}$ such that $$\mathbb{E}_{{\bf{\epsilon}}}[{\bf{\epsilon}}] = 0$$ I want to find the expectation of the inner product between another vector which is ...

1 2 3 4 5 33