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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RBF transformation on a Normally Distributed Random Variable

I have a random vector $\mathbf{X} \sim \mathcal{N}(\mathbf{m,\Sigma})$ which is transformed by a Gaussian Radial Basis Function into the random variable $\mathbf{Y} = K(\mathbf X) = \exp(-\lambda ...
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10 views

Realistic simulation of instrumental data

I want to simulate data collected by an instrument realistically. The problem is that if I use simple Gaussian deviates then an analysis program can guess the truth simply by computing the mean of a ...
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49 views

Is the ratio distribution of two normally distributed variables ever normal?

Let $Z = X / Y$ where $X$ and $Y$ are normally distributed random variables. Is $Z$ normally distributed for any $X$ and $Y$?
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46 views

If $X$ and $Y$ are normally distributed random variables, what kind of distribution their sum follows?

I was reading this question. It is about notation but I would like to ask something regarding the sum of two normally distributed random variables. If $X$ is a normally distributed random variable ...
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1answer
59 views

About the $X\sim \mathcal{N}$ notation

If a random variable $X$ has mean $\mu_{X}$ and variance $\sigma_{X}^{2}$, and follows a normal distribution, it may be written as $X\sim \mathcal{N}(\mu_{X},\sigma_{X}^{2})$. Suppose $Y$ is also ...
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Find a connection between 2 variables when not all variables are known

Lets say you have a system which contains two variables X and Y. You know they are connected but you don't know how. You also know that the system most likely has other variables which effects it, but ...
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25 views

Does a random variable have to be a total function on the space of outcomes?

A random variable is a function from the space of outcomes to real numbers (there are extensions to this, but it's not important for the purposes of this question: see wikipedia). The question is: ...
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19 views

Strictly positive random variables

Suppose $X\sim N(\mu, \sigma^{2})$ with some small $\sigma^{2}$ and largish $\mu$. Now $X$ will be rarely negative. Suppose I need random variables that are strictly positive but otherwise ...
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111 views

What is the expected value of $\frac{X}{X+Y}$?

I am trying to find the expected value of $\displaystyle E\Bigg[\frac{X}{X+Y}\Bigg]$. I started with writing $\displaystyle E\Bigg[\frac{X}{X+Y}\Bigg] = ...
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1answer
37 views

Variance of the $\hat{\sigma^2}$ of a Maximum Likelihood estimator

Given some normally distributed observations $x_1,x_2,...,x_n$ $\forall i\ x_i\sim\mathcal{N}(\mu, \sigma^2)$ the ML estimator decides that the variance that maximizes the likelihood function is ...
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1answer
46 views

How to calculate the probability distribution F(X,Y) when the distributions of X and Y are known?

Suppose $X$ and $Y$ are normally distributed with known means and standard deviations. How do I calculate what is the probability distribution some function $f(X,Y)$? For example, say $f(X,Y)=2X+Y$. ...
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15 views

generate multivariate cauchy random variable

How can we generate multivariate random variable in matlab? Will the set of t distribution with 1 degree of freedom will give multivairate cauchy noise, How to generate?
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1answer
35 views

Probabilities of conditional expectation values in uniform distribution

Let's consider a continuous random variable $X$ as follows: $f_X(x)=\left\{ \begin{array}{ll}\frac{1}{2}, &\mbox{if} \ x\in[0,1] \\ \frac{1}{4}, &\mbox{if}\ x\in(1,3]\end{array}\right.$ ...
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15 views

General definition for higher order co-moment matrix

Is there a general equation or procedure for computing higher-order co-moment matrices (i.e., coskewness matrix, cokurtosis matrix, etc) for a vector of random variables? For example, the covariance ...
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1answer
109 views

How to calculate probability of observing a value given a permutation distribution?

I have a single observation with value $x = 0.5$ that comes out from a complicated computational process. I would like to know what is the probability to observe such value by chance. To attempt to ...
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2answers
40 views

Observed Vs Unobserved Variables [duplicate]

Can anyone explain the difference between observed variables and unobserved variables (preferably in plain English ) ?
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46 views

4 cases of Maximum Likelihood Estimation of Gaussian distribution parameters

Let $x_1,x_2,...,x_n$ some normally distributed observations. So $\vec{x}=\begin{bmatrix}x_1 & x_2 & ... & x_n\end{bmatrix}^{T}$ In the context of my research I am trying to estimate ...
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1answer
50 views

$E_{\theta_1}[\ell(\theta_2;X)] $

I am faced with the following in my "Statistical inference book" $E_{\theta_1}[\ell(\theta_2;X)] $ where $\ell(\theta_2;X)$ (loglikelihood) is $\log [P(\theta_2;X)]$, X is a random variable. What ...
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22 views

3 correlated questions: Find the roots of an equation, find the inverse of a function and find the c.d.f. of a function of a random variable

I have 3 correlated questions: Find the roots of an equation, find the inverse of a function and find the c.d.f. of a function of a random variable. The questions are in the picture. Sorry for the bad ...
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26 views

How to calculate an error estimate for a sum of random variables when you only know several “subsums” of the variables?

I have a sample of $N$ iid random variables and I would like to get an error estimate for the sum of the variables of my sample. However I have very limited knowledge about my sample: I only have $k$ ...
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21 views

What are the statistical properties of time delayed lagged time series?

Performing Taken's phase space delay embedding on the observations $\mathbf{z}$ of a univariate random variable, with an embedding dimension $d$, we get a realization of $n$ points such as: ...
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1answer
30 views

Time series and random variable

I would like to know if the $n$ realizations of a variable, say $Y$ expressed in the form of a time series constitutes $n$ random variables or just a single random variable $Y$? For example, the ...
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1answer
69 views

Example of singularly continuous random variable

I am looking for a useful example of a continuous but not absolutely continuous random variable, that is for which the cumulative distribution function is not differentiable. By useful example, I ...
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1answer
34 views

Mixed Effects model with block and autocorrelated fixed effect

I am trying to run a mixed effect model on soils data that were collected at two locations in a randomized complete block design (4 blocks). I am interested in the effect of location and depth (below ...
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74 views

Expectation of quotient of linear combinations of independent standard normal random variables

Let $a, b, c, d, e, f$ be complex numbers with nonnegative real parts and nonnegative imaginary parts, and let $X_{1}, X_{2}, X_{3}, X_{4}$ be independent standard normal random variables. How can I ...
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Relationship between the Gamma and Beta distributions

I was looking at a proof of the following fact Let $X \sim \mbox{Gamma}(\alpha, 1)$ and $Y \sim \mbox{Gamma}(\beta, 1)$ where the paramaterization is such that $\alpha$ is the shape parameter. Then ...
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1answer
40 views

Proof for the p.d.f of minimum and maximum of a sample

The following is a question from a past paper for one of my university statistical inference modules, and I know how to use the formula for each the max/min, but Assume that the sample $X_1, X_2, ...
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15 views

Bayesian posterior with multiple signals and constraining equation/slice

Prior and signals: Let $y_1$ and $y_2$ be iid signals on $Y$. The intial prior is $Y \sim N(\bar{Y}, \sigma^2_Y)$, where $N(\cdot, \cdot)$ is the normal distribution The signals are independent and ...
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21 views

Definition of random variable [duplicate]

In some books, they don't define the random variable based on measure theory. Instead, they define as follow (in the book All of Statistics of Larry Wasserman): My question is does this definition ...
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78 views

Mixed distribution of product of Bernoulli and Gaussian r.v

What should be the density function of the following mixed distributed random variable $Z$. $$Z \equiv X \cdot Y,$$ where $( \cdot)$ is product operation, and $X$ and $Y$ being independent, $X$ is ...
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1answer
68 views

R, Pairwise comparison of random variable in mixed model

We measured temperatures of a pond repeatedly every day at each hour for a month at two different depths (i.e., top and bottom). We want to see if the temperatures at the top of the pond are ...
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2answers
48 views

Square of gamma random variable [duplicate]

If i have a random variable with distribution $X \sim \Gamma(\alpha,\beta)$ then what would be the distribution of $Y = \lambda X^2$ (with $\lambda$ a scaling factor)? Can I say that $Y$ will follow a ...
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1answer
71 views

Inferring parent distribution using phase of 2D discrete fourier transform of random image

Suppose I have an NxM matrix with real entries $S_{n,m}$ which are iid with zero mean. Say $n \in [0, 1, ..., N-1], m \in [0, 1, ..., M-1]$. From $S_{n,m}$ I can compute the 2D discrete fourier ...
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1answer
50 views

Variance stabilisation

$Y$ has mean $\mu$ and variance function $V(\mu)$. If $V(\mu) = \alpha.\mu^v$ then $h(y) = y^{(2-v)/2}$ is variance stabilising which means that $Var(h(Y))$ is approximately constant. I tried to ...
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31 views

Advice on ANOVA formula for nested random design

I have read "Design and Analysis of Experiments" 8th Edition by D.C. Montagomery. In Chapter 14, there is a nested experiment with two factor A and B. Both factors are random. Then the ANOVA test ...
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1answer
64 views

Estimate variance of a function given variances of variables

I'm given the mean and the standard deviation of N random variables $A$, $B$, $C$, $D$... I compute the function $f:=f(A, B, C, D...) = \frac { AB... }{ CD... } $. How can I estimate the variance of ...
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196 views

Is it feasible to transform each variable differently while doing multiple regression

I have a dataset with 10 variables ...is it feasible to transform each variable differently while doing multiple regression... for example new_V1 = log(v1) New_V2= V2^2 New_V3= 1/V3 Likewise ...
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61 views

Bivariate distribution: beta and binomial

Consider a pair of RVs $X$ and $Y$, with the following conditional distributions: $$X | Y=y \sim Binom(L, y)$$ $$Y | X=x \sim Beta(\alpha + x, \nu)$$ where $L$, $\alpha$, and $\nu$; are all ...
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310 views

Using Uniform Distribution to Generate Correlated Random Samples in R

[On recent questions I was looking into generating random vectors in R, and I wanted to share that "research" as an independent Q&A on a specific point.] Generating random data with correlation ...
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2answers
63 views

Variance of the linear transformation of a random variable

I have a problem where the variance I'm calculating does not seem right. I have the following data: ...
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3answers
95 views

Generating and Working with Random Vectors in R

I've been trying to understand random vectors and generate them in R to reproduce properties. Recently, I asked a similar question, and it was rightfully placed on-hold for being too general. Thanks ...
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1answer
32 views

Which of the following when performing a statistical test is NOT a Random Variable?

For a test for my stats class, we were asked, "When performing a statistical test on a sample, which of the following is NOT a random variable?" A) The Test Statistic B) the p-value C) The power of ...
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21 views

Cointegration of random processes

If we assume that two random processes are cointegrated, do we implicitly make an assumption about the way that the two random processes are generated?
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4answers
166 views

Why is $\mathbb E(X)=\sum_{i=1}^{n}x_i P(x_i)$?

If $X$ is a random variable and $x$'s are the realizations form $X$ and $N$ is the population size $n$ is the sample size Which one is correct $\mathbb E(X)=\sum_{i=1}^{N}x_i P(x_i)$ or ...
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31 views

Random Walk Probability Including Drift

What is the equation for the probability of a random walk with drift being equal to a specific value after n steps, given a specific standard deviation?
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171 views

What is a multivariate random variable?

I've been trying to read the Wikipedia article on multivariate random variables but I'm having trouble getting past the math. Is there a more intuitive explanation? I'm assuming that a univariate ...
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1answer
44 views

How verify if a dataset is IID

I have a dataset of 100000 samples. The samples represent the failure time of an electronic component that fails after a given number of "shocks", whatever shock means. We know that these systems fail ...
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1answer
114 views

Finding the support of transformations of random variables

Let $X, Y \sim iid U(0,1)$ and $c_1, c_2 \in \mathbb{R}$. In the linear combination $Z = c_1X+c_2Y$, we know that the probability density function of $Z$ depends on the relationships of $c_1$ and ...
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Simulating Random Vector X in R to Confirm with Example that cov(AX) = A cov(X) A'

As has been answered previously, the proof of cov(AX) = A cov(X) A' with X being a random vector and ...