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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Function of random variable

If I have a random variable $X$ which has mean $\mu$ and variance $\sigma^2$, what is the approximate expression of $log(X)$ and $\sqrt{X}$? Do I assume normal approximation or use Taylor expansion?
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42 views

showing a random variable has an exponential distribution

Let $X_{1},..,X_{n}$ be independent, each with a exp($\lambda$) distribution. Let $Z=min(X_{1},..X_n)$. Show that $n\lambda Z$ has an exp$(1)$ distribution. I calculate that $P(Z>z)=e^{-n\lambda ...
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2answers
131 views

Choosing a discrete non-uniform distribution for generating random integers

I have a list $l$ containing integers in the range $[1,max]$ On list $l$ I do an operation $isPresent(x)$ which return true if ...
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0answers
27 views

Deconvolution of sum results in negative numbers

Given $T=G+A$ where $A$ and $G$ are independent random variables, I'd like to estimate the distribution of $G$ given empirical (measured) distributions of $T$ and $A$. Of note: all three random ...
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1answer
25 views

tight bound of bernoulli sums with unknown dependency

Consider n random variables $X_1, \ldots, X_n$ all follow same bernoulli distribution of mean $p$. But the dependency of these variables are unknown (i.e., cannot assume that they are independent). ...
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1answer
42 views

Simulating Diffusion/Wiener Process with Random Walk [closed]

I hope this is the right section for this kind of questions. I am trying to simulate, with MATLAB, a diffusion model starting from a Random Walk. I am using a Random Walk with information increment ...
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17 views
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1answer
17 views

decomposition of independent random variables

If X and Y are independent random variables such that X = A + B and Y = C + D? Are the pairs (A, C), (B, C), (A, D), (B, D) also independent? By this I mean whether A and C are independent, B and C ...
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1answer
22 views

Multiplicative error in survey data

I'm working on a panel survey data where each individual's income was multiplied by a individual-specific random number (each random number is evenly distributed from 0.5 to 1.5) to avoid any ...
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0answers
12 views

Minimal I-maps induced by sets of scopes. Clarification needed

I have a question about Prop. 9.1 on page 307 in "Probabilistic Graphical Models" (link to google books) (Koller / Friedman). I don't see why $\mathcal{H}_{\Phi}$ is a minimal I-map: If I have only ...
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2answers
315 views

How can stochastic gradient descent avoid the problem of a local minimum?

I know that stochastic gradient descent has random behavior, but I don't know why. Is there any explanation about this?
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1answer
25 views

Consequences of identical distribution of random variables

Consider two real-valued random variables $Y_1: \Omega \rightarrow \mathbb{R}$ and $Y_2:\Omega \rightarrow \mathbb{R}$. Even if they have the same domain and codomain, $Y_1 $ and $Y_2$ generate ...
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1answer
40 views

Dealing with guessing in multiple choice questionnaires

I just read this evaluation of a multiple-choice test. The second question of the test is a simple yes/no question which can obviously be guessed correctly with a certain probability. The analysis has ...
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35 views

Understanding a double peaked distribution

I am running a Monte Carlo Simulation and am sampling randomly from about 65 Normal Distributions, each with a different $\mu$ and $\sigma$. I end up with the Mixture Distribution graph shown below ...
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0answers
37 views

What is the minimum of two independent variables?

I am having trouble to understand the min {X,Y} where X and Y are independent random variables. From the online source, it says that min{X,Y} < x if and only if X>x, Y>y. First of all, what ...
4
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1answer
81 views

Method for simulation for random variables with a particular probability density function

Suppose one has an expression for the probability density of a random variable- how does one simulate for the particular random variable. I understand that this isn't an issue for probability density ...
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1answer
48 views

Can I use the Cholesky-method for generating correlated random variables with given mean?

I want to generate correlated random variables with a given correlation matrix, means, and variances. Does the Cholesky decomposition only work when the initial random variables are iids with the same ...
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21 views

Correcting for sampling in arbitrary distributions

Let's say I've got a series of Gaussian random variables $^{1}x_{i} \sim N(x_i,\sigma_i)$, with each $\sigma_i$ different, but known a priori. I've got another identical series of Gaussian random ...
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40 views

A simple mathematics?

I am struggling on how to put below problem with simple mathematics. Let say I have 2 variables N & p. The 2 instances of N are N1 & N2 and 2 instances of p are p1 and p2. Now please ...
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22 views

Hausman test with or without covariates

I am using panel data and I would like to determine whether I can use the Random Effect (RE) model instead of Fixed Effect (FE) to estimate one coefficient of interest. When I use the Hausman test ...
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1answer
30 views

How to sum uncertainties, systematic and random

I apologize for the simplistic questions. I have a retrieval process that has a set of random and systematic uncertainties associated with it. I'm assuming that these are all independent. The goal is ...
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17 views

Assuming a raster image is a random vector

I am currently working in remote sensing. It is quite common to find that the beginning of the statistical descriptions of techniques used in satellite image processing begin with defining the initial ...
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1answer
30 views

Distribution of a simple random sample

Homework Question Hello, I have a simple homework question that asks: We know that when we sample, with equal probability from a finite population {x1, x2,..., xN} without replacement, we obtain a ...
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1answer
18 views

I have a source of numbers. Is there a way to estimate the probability for the given number to originate from it?

The source in question is some blackbox which output looks completely random to me, yey I haven't tested if for randomness quality. I don't exactly know insides of blackbox but can easily caputure it ...
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36 views

repeated measure ANOVA or GLMM?

I would like to ask for some advice on an analysis that I am doing at the moment. The experiment: I ran an experiment with animals in which I subjected 15 pairs of individuals (one male and one ...
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1answer
111 views

What is the “equivalent” of normal distribution in an interval?

What is the most "natural" family of distributions on an interval [0,1] indexed by their mean $\mu$ and standard deviation $\sigma$? I am looking for something occurring in "nature", like normal ...
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91 views

Estimating variance from sequence of random variable

Given $X\sim N(0,\Omega)$. Suppose that we can construct a sequence $\{X_n\}$ based on the observation such that $\{X_n\}\to X$ in distribution. My problem is to estimate $\Omega$ consistently using ...
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1answer
75 views

prove that this expression converges in probability to zero

Apparently $T^{-3/2} \sum\limits_{t=1}^T{y_{t-1}}u_t$ converges by law to $0.5\times T^{-1/2}\sigma^2 (X-1)$, where $X$ is a $\chi^2(1)$ random variable. $u_t$ is white noise and $y_t$ is an AR(1) ...
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85 views

How to generate Normal variables parts of which are correlated (in R)?

My second attempt to explain the question. Start with a vector of numbers V1 of length M. The elements of V1 form a Normal distribution. Take (any) N elements from this vector. Generate a replica of ...
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34 views

The distribution of the product of two random variables

X=Normal(0,1) random var. Y=Uniform{-1,1} random var. Show that Z=X.Y is normal random variable. Thanks for your help in advance. I tried to solve it through double integral but it failed to do. ...
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103 views

What is the expectation of one random variable divided over another (both independent)?

Suppose I have $X,Y$, which are independent random variables. Why is it that $E(\frac{X}{Y}) = E(X)E(\frac{1}{Y})$? Also, why is it that $E(X^2Y^2)=E(X^2)E(Y^2)$? How is it that the square of an ...
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3answers
325 views

Is the “test statistic” a value or a random variable?

I am a student taking my first Statistics course now. I am confused by the term "test statistic". In the following (I saw this in some textbooks), $t$ seems to be a specific value calculated from a ...
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3answers
139 views

Simplest example of uncorrelated but not independent X and Y? [duplicate]

If any hard-working student is the simplest counterexample to "all students are lazy", what is the simplest counterexample to "if random variables X and Y are uncorrelated they are independent"?
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Is this statement about the conditional expectation of a sum true?

For expectations of random variables (RVs) $X$ and $Y$ it is true that $$E(X+Y)=E(X)+E(Y)$$. My question is whether when conditioning on RV vector $Z_{1...J}$, it is also true that ...
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1answer
39 views

Deviate vs. Variable

What is the difference between a multivariate normal random deviate and a random variable? More specifically, I suppose, what exactly is a deviate?
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1answer
70 views

Sufficiency of order statistics

I am told the following proof is incorrect, but I cannot understand why. Consider $X_{(1)}, \ldots, X_{(n)}$ are the order statistics of a random sample of size $n$. I want to show that the order ...
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75 views

maximum gap between order statistics of normally distributed random variables

Hello Cross Validated community, I am currently working on a not-that-easy problem involving order statistics. As I am unsure as to how I could solve it, I thought it might already possess a ...
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31 views

Central Limit Theorem: Likelihood multiplication

Been watching this video by Tom Minka on Expectation Propagation (http://videolectures.net/mlss09uk_minka_ai/). At about 19:12, he says that the reason the moment matching technique works is that when ...
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1answer
31 views

Expected value of coin tossed twice

Here's the problem: A (Laplace) coin with sides $0$ and $1$ is tossed twice, each time independently from the other. Let $f:=$ maximum of both results $g:=$ sum of both results. What is the ...
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1answer
42 views

Is there a way to measure the viability of an experiment?

Is there a way to calculate a number that evaluates the viability of an experiment? By that I mean the probability that the observed probability is just chance. I read about the $p$ value, but it ...
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1answer
80 views

Notation for Random Bernoulli-Like Vector With Fixed Sum

I draw a random vector of dimensionality $k$, each dichotomous element of which taking on a value in $\{0,1\}$. The probability that any element will be $1$ is captured in the $k$-dimensional ...
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1answer
83 views

how calculate expected value

(Ross [2009], p.162) The current in a semiconductor diode is often measured by the Shockley equation I = I0(e^aV-1) where V is the voltage across the diode; I0 is the reverse current; a is a constant; ...
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1answer
98 views

Which Venn diagram is appropriate here for statistically independent, uncorrelated and orthogonal random variables?

I understand concepts more with visualizations. So I made a Venn diagram for statistically independent, uncorrelated and orthogonal random variables. But I am in a confusion which of the below Venn ...
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1answer
62 views

Variance of a sum of functions of random variable

I'm trying to prove $$ \mathrm{Var}\left(\sum\limits_{i=1}^n{g(X_i)}\right) = n(Var(g(X_1))) $$ where $X_1...X_n$ are IID variables. I have been trying to use proof for a similar question - ...
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1answer
33 views

$X_{n}$ and $Z$ be random variables if $X_{n} \ge Z$ then $ E[\liminf_{n\to \infty} X_{n}] \le \liminf_{n\to \infty} E[X_{n}] $

Let $X_{n}$ and $Z$ be random variables on probability space $(\Omega ,\mathcal F,P)$ and $Z$ be integrable. show that $$X_{n} \ge Z \qquad \Longrightarrow \qquad E[\liminf_{n\to \infty} X_{n}] \le ...
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44 views

I want to show $Y_{n} \xrightarrow{ p} c$ ,then show that $f(Y_{n}) \xrightarrow{ p} f(c) $

Let $c \in \mathbb R$ . Let be a sequence of random variables . let $f:\mathbb R \to\mathbb R$ be continous function . if $Y_{n} \xrightarrow{ p} c$ ,then show that $f(Y_{n}) \xrightarrow{ p} f(c) $
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1answer
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Meaning of covariance matrix row sums

Say I have an $n \times n$ covariance matrix for a sample set of $n$ random variables. Is there any meaning if the sum of the rows of this matrix? Is it a meaningful measurement of the contribution ...
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1k views

Inverting the Fourier Transform for a Fisher distribution

The characteristic function of the Fisher$(1,\alpha)$ distribution is: $$C(t)=\frac{\Gamma \left(\frac{\alpha +1}{2}\right) U\left(\frac{1}{2},1-\frac{\alpha }{2},-i t \alpha \right)}{\Gamma ...
3
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4answers
208 views

Spherical platykurtic random cloud

I'd like to generate a multivariate continuous data which is globular cloud, like multivariate standard normal data is, but which is more platykurtic than normal data. There are many ways to get ...
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20 views

Integrability of a sequence of iid random variables

I'd really appreciate some hints on the first part of the following question: Let $f_n, n\in \mathbb{N}$ be a sequence of iid random variables over $(\Omega, A,P)$. That is, $P(\{f_1 \in ...