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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Which groups are significantly different from within a random effect (using lmer in R)

I am using a mixed effects model as created here (using dummy data for now) in this R script ...
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65 views

Calculate $\mathbb{E}[Z_i]$ where $Z_i = \min(X_i, Y_{i-1})$, $X \sim Beta(\alpha,1)$

Let $X$ be a IID random variable with support in $[0,1]$ and CDF given by a Beta distribution, i.e. $X \sim \mathrm{Beta}(\alpha,1)$. Let $Z_i = \min(X_i,Y_{i-1})$, $\forall i >1$. I would like to ...
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39 views

Sum of Random Variables

As part of my statistical mechanics class, I'm trying to go through Kardar's statistical physics of particles and I'm having trouble with this one line: Consider the sum $X=\displaystyle ...
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2answers
61 views

Skewness of a random variable that have zero variance and zero third central moment

If I have a random variable $x$, and the only information I know about it are: $$ m_1=E[x]=c, \mu_2=var(x)=0, \mu_3=E[(x-m_1)^3]=0$$ Can I conclude that the function distribution is symmetric about c? ...
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1answer
36 views

How is this Negative Binomial Random variable used to solve this problem?

I was looking at the solution to this problem below and I don't understand how they used a negative binomial R.V. to solve the problem. A research study is concerned with the side effects of a new ...
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1answer
143 views

Do mean, variance and median exist for a continuous random variable with continuous PDF over the real axis and a well defined CDF?

For a continuous random variable with continuous PDF over the real axis and well defined CDF, are the mean, variance, and median always well defined? Mean and variance do not always exist, e.g. for a ...
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1answer
47 views

Expected Value of Random Variable

I'm trying to find the expected value of a random variable $t_i$ which is the solution of $$\epsilon_i=\mu(t_i-t_{i-1})-\sum^{i-1}_{k=1}\frac{\alpha}{\beta}\left(1-e^{-\beta(t_i-t_k)}\right)$$ ...
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1answer
32 views

Does the average of the square roots of random variables mean anything?

I recently made a plot for work that used a signed square-root scale on the $y$ axis, for visual clarity. The $y$ observations are impulse response functions (IRF) of vector autoregressions computed ...
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9 views

probability life similarities in a population cross section

I have just read up on Bouchard's Minnesota Twins study that turned up some amazing identical similarities in the lives of some identical twins reared apart. Both had same jobs, married and divorced ...
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1answer
26 views

Random variable variance

I have the model $y_i=\beta_1+\beta_2 X_i+ u_i$ where $u_i\sim\text{iid } N(0,\sigma^2)$. I estimate $\beta_1$ and $\beta_2$ by drawing a straight line between the first $(x_1,y_1)$ and last dot ...
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+50

Applying a variance-stabilizing transform to a fitted function (rather than data)

Outline I'm working with data corrupted by a mixed Poisson-Gaussian noise model (for example with images gathered in astronomy or electron microscopy), and have been using the generalized Anscombe ...
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18 views

Estimating number of intersections between point and polygons

I have a 2D plane (a large rectangle) with a finite size in the x and y direction, which is the field of my problem. The field is covered by $n$ smaller rectangles that are located randomly within ...
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3answers
64 views

What is the correct notation if two random variables belong to the same distribution?

I want to explain that both the real and imaginary part of a complex variable follow a zero-mean complex Gaussian distribution. How can I write that? For one variable I'd write $a \sim ...
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41 views

probability that a variable is ONE OF the top k out of n when ordered

Suppose ($h_1,h_2,...,h_n$) is an $n\times 1$ vector. Let $h_i=g_iX_i$, where $g_i$ is a non-random variable which can vary across $i$ and $X_i$ is a random variable with Pareto Type I distribution. ...
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3answers
71 views

Biased coin toss simulation — which random generator is most appropriate?

I have a doubt whether the uniformly distributed random generator is the most appropriate to use for simulating biased coin toss. ...
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4answers
41 views

Conditional variance - $Var(X + U | X) = Var(U)$?

I am wondering if the following equality holds - $Var(X + U | X) = Var(U)$? where $X$ and $U$ are two independent random variables? It seems can we say $Var(X + U | X) = Var(X|X) + Var(U|X) = ...
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1answer
25 views

Notation: Deterministic Variable, Random Variable, Realization of Random Variable, Function

Is there a standard notation for distinguishing between the following: Random Variables Realizations of Random Variables Deterministic Variables (not random) Functions I am familiar with using ...
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26 views

How to apply the random statement in my model?

i am trying to model the following data in SAS using proc glimmix and i would like feedback if i have modelled my data correctly. My data consist of individual chickens (ID) who are grouped by ...
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2answers
39 views

Probability generating function for negative values of random variables?

What if we have negative integral values for a random variable?Then is it possible to write a probability generating function for it? All definitions I have seen so far is for non negative integer ...
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1answer
21 views

Laplace transform and density

It is true that the Laplace transform of a (positive) random variable characterises that random variable, just like its density? ($L_X(z) = E(exp(-Xz))$)
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31 views

Largest eigenvalue of this random 2x2 matrix

Consider four random complex vectors $\mu_i$ of length $K$ whose entries are drawn from the complex normal distribution $\mathcal{CN}(\mathbf{0},\mathbb{1})$ centered in zero and of unit variance. ...
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20 views

Measure of probability of obtaining a certain $\chi^2$ given the number of trials

I am calculating an equation with several random components a large number of times, and in order to select my final answer I choose that with the minimum value for $\chi^2$. I am hoping to find some ...
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1answer
94 views

Squared random variable $X^2$ vs $X\times X$

As I understand a random variable represents all possible outcomes of an experiment with their associated probabilities. Why $X^2$ is understood as squaring outcomes of experiments instead of as ...
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1answer
114 views

If a random variable V is independent of two independent random variables X and Y, how to prove that V is independent of X + Y?

This is question 3.8.4 of An Introduction to Mathematical Statistics and Its Applications, 5th Edition, by Larsen and Marx. This is not homework for a class I am taking now, but might someday be for ...
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1answer
42 views

correlation of two sums of random variables

Imagine two random variables $X$ and $Y$ which are correlated with $\rho = 1$. Both have a mean of $100$ and a standard deviation of $40$. Two other random variables $U$ and $V$ are correlated at ...
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1answer
28 views

Dirichlet sample by normalising Gamma RVs

I know that if you sample $K$ random variables $(X_1, X_2, \dots, X_K)$ from Gamma distributions using shape parameters $(\alpha_1, \alpha_2, \dots \alpha_K)$ and a scale parameter $\theta = 1$ such ...
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1answer
50 views

Rectifier function of random variable

Let $X$ be a random variable with distribution $F_X$ and density $f_X$. Define $$g(x) = \left\{ \begin{array}{lr} x & : x \ge 0\\ 0 & : x < 0 \end{array} \right\}$$ and let ...
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1answer
69 views

Probability function of three Random Variables multiplied, solidifying intuition

I ran across this exercise: Let $T$ be a random variable distributed as a $\text{Bernoulli}(p)$, $U$ be a random variable distributed as a $\text{Bernoulli}(q)$ and $W$ be a random variable ...
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1answer
60 views

Random variable as a function

I'm reading Schaum's outline of probability, random var. and random processes. In the second chapter they make it clear that a random variable $X$ is not a variable in the traditional sense but ...
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56 views

Markov Chain exercise in an exam

Suppose that $X_1, X_2, X_3... $ is a Markov chain with the following transition matrix: State: | 1 2 3 --+----------- 1 |0.2 0.4 0.4 2 |0.5 0.0 0.5 3 |0.6 0.3 0.1 (forgive my attempt at a ...
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14 views

Choosing levels of random factor in a mixed model

My apologies in advance if this question is as much about experimental design as it is about statistics. I'd like to know if there is a "best" way to choose levels of a random factor in a mixed model. ...
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1answer
57 views

Are explanatory variables considered random in PCA?

One of properties of PCA states that the sum of the variances of the principal components is equal to the sum of the variances of the explanatory variables. I wonder how to interpret this as I've ...
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10 views

Mutual information decrease with coarse-graining

Let $X,A,Y,B,C,D$ be random binary variables. $D$ is independent from $X,A,C$ and $C$ is independent from $Y,B,D$. Is it true that: If $I(Y:B|D=0)\leq \epsilon$ then $I(X\oplus Y:A\oplus ...
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37 views

Random variables: True or False

I sometimes get confused when reading statistical definitions when they mention random variables (RV). Are they talking about a single draw? Are they talking about an estimator? To make things clear ...
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1answer
33 views

Decomposing a Combination of Distributions?

Let's say I'm studying the distribution/log-distribution of a random variable that may actually be the result of a combination of distributions. What are the ways I can check for this and possibly ...
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30 views

Average of several categorical random variables

Say I have a categorical random variable $X$ over a discrete label space $L$ = {Sky, Road, Tree, Unknown}. In other words each $X_i \in L$. Now I store the parameters of each $X_i$ which are the ...
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47 views

Calculating random effects from a glmerMod object (r package lme4)

Using the lme4 package, how does ranef() calculate (or extract) estimates of random effects from a ...
3
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1answer
48 views

Confidence interval for the “shift parameter” of a non-central exponential distribution

Suppose that $X_1, X_2, \ldots, X_n$ (with $n > 0$) is a random sample from a non-central exponential distribution with probability density function: $$f(x | \lambda, k) = \lambda * ...
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1answer
59 views

Maximum likelihood estimation for a sequence of observations

Assuming $A_1, A_2, \ldots, A_n$ are independent exponential random variables (each having the same parameter and, for the sake of simplicity, let's assume the value of the parameter is 1). Define ...
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1answer
70 views

Probability density function of the sample maximum of a random variable

According to my book, for a random sample $(X_1, \ldots, X_n)$ from a continuous distribution with p.d.f. $f(x)$ and c.d.f. $F(x)$, the p.d.f. of the maximum of the sample is $g(z)=nf(z)[F(z)]^{n-1}$, ...
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45 views

Quantifying variable importance for GLMM using hierarchical partitioning (in R)

I am interested in quantifying variable importance for a binomial logistic mixed-model regression. My model has 5 fixed effects, and 3 random effects (2 nested). I am doing model selection and ...
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1answer
31 views

How to decompose the probability of sum of a series random variables?

Let $A_1, A_2, \ldots$ be independent random variables with known distributions. I'm wondering if it's possible to equivalently decompose the probability above into separated constraints? For ...
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1answer
53 views

Difference of two independent gamma distribution

Given two independent random variables $X\sim\Gamma(s,r)$ and $Y\sim\Gamma(t,u)$, what is the distribution of the difference, i.e. $D=X−Y$? I assume that $s$ and $t$ are integers. How can I obtain the ...
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1answer
86 views

Probability distribution of functions of random variables?

I have a very stupid doubt: consider the real valued random variables $X$ and $Z$ both defined on the probability space $(\Omega, \mathcal{F},\mathbb{P})$. Let $Y:= g(X,Z)$, where $g(\cdot)$ is a ...
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23 views

A binomial random number generating algorithm that works when n*p is very small

I need to generate binomial random numbers: For example, consider binomial random numbers. A binomial random number is the number of heads in N tosses of a coin with probability p of a heads ...
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1answer
38 views

How do I prove the mutual independence of a set of random variables given another set of mutually independent random variables?

Suppose $X_{1},\,\ldots,\, X_{n}$ and $T_{1},\,\ldots,\, T_{r}$ and two sets of random variables, with each $X_{i}$ being categorical, and each $T_{j}$ being continuous taking on all values in the ...
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4answers
804 views

Can anyone clarify the concept of a “sum of random variables”

In my probability class the terms "sums of random variables" is constantly used. However, I'm stuck on what exactly that means? Are we talking about the sum of a bunch of realizations from a random ...
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2answers
63 views

Convergence in distribution results not deriving from central limit theorem?

In the stats class I took, all the results I have encountered about the convergence in distribution of some random variables are in one way or another consequences of the Central Limit Theorem. Out ...
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1answer
64 views

Random Variable

Three components are randomly sampled, one at a time, from a large lot. As each component is selected, it is tested. If it passes the test, a success (S) occurs; if it fails the test, a failure (F) ...
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325 views

Functions of Independent Random Variables

Is the claim that functions of independent random variables are themselves independent, true? I have seen that result often used implicitly in some proofs, for example in the proof of independence ...