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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Distribution of the exponential of an exponential random variable

Let $X$ be a real valued random variable with exponential distribution. Let $a$ be a complex number. What is the distribution of $Y = e^{aX}$? Can Y be written in the form of another known ...
11
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1answer
343 views

Why is a “negative binomial” random variable called that?

I don't understand why the "negative binomial" random variable has that name. What is negative about it? What is binomial about it? What is negative-binomial about it?
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22 views

Can any distribution be fully described by the [infinite] set of all its moments? $\{E[x^n]\}_{n\in N}$ [duplicate]

Is it possible describe any distribution uniquely by the infinite set of all it's moments $\{E[x^n]\}_{n\in N}$ If yes, does that include discrete, truncated, etc. distributions? If not, is this ...
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2answers
86 views

What does it mean 'A random variable is a function'?

I have sometimes seen the formulas such that $\operatorname{Var}(aX+b)=a^2\operatorname{Var} X$ and $EX^2=\sum_{x=0}^nx^2\binom{n}{x}p^x(1-p)^{n-x}$. But how do we define $aX+b$ and $X^2$ if, say ...
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0answers
23 views

probability of random sample mean

company x - mean life time: 6.5 variance: 0.81 company y - mean life time: 6.0 variance: 0.64 what is the probability that a random sample of 36 from company A will have a mean life time that is at ...
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1answer
36 views

Existence of expected value of function

According to Wikipedia, the expected value of a continuous random variable is $$E[X] = \int_{-\infty}^{\infty} xf(x) \mathrm{d}x.$$ Suppose $f$ is a function such as $f:\mathbb{R}\rightarrow ...
2
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42 views

MI between two random vectors

$ \vec X=\begin{bmatrix} X_{1} \\ X_{2} \\ \end{bmatrix} $ and $ \vec Y=\begin{bmatrix} Y_{1} \\ Y_{2} \\ Y_{3} \\ \end{bmatrix} $ ...
0
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1answer
50 views

Notation and explanation for certain conditional random variables

This is a two part question. I apologize if the title and tags are vague. Please edit if a more suitable title or tags are appropriate. Part 1 Ok, so if $X$ and $Y$ are independent, continuous ...
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1answer
91 views

Test if two normally distributed random variables have the same mean

We have two independent random variables which follow normal distributions $X_1\sim \mathcal N(\mu_1,\sigma_1)$, $X_2\sim \mathcal N(\mu_2,\sigma_2)$. From the context, we have that $\mu_1\leq\mu_2$. ...
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1answer
23 views

Independence betweeen sets of random variables

Given two random variables $A$ and $B$, I know we can call them independent if their joint PDF is factorizable to a product of their marginal PDFs: $f_{A,B}\left(A,B\right)=f_{A}\left(A\right)\cdot ...
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0answers
13 views

Argmax of transformed RV

Does performing a (nonlinear) transformation to a random variable, finding the value in that space that maximizes the PDF, and transforming that value back to the original space yield the same maximum ...
1
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1answer
66 views

Notation for constant random variables?

Suppose $X$ is a random variable. Now, suppose I want to add a constant random variable to $X$. Should I denote the constant by a lower- or upper-case letter? So $X+A$ or $X+a$?
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83 views

What is the simplified expression for autocorrelation in this case?

The CRB gives the variance of the estimation error of the estimates and a lower value is preferred. I have computed the cramer rao bound (CRB) of the estimates of the coefficients $\mathbf{h^T}$ for ...
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1answer
47 views

Not all positive-definite matrices are valid covariance matrices for lognormal variables

A simple method to generate correlated lognormal variables $X_i$ that obey a covariance matrix $C_{\mathrm{ln}}$ with elements $c_{\mathrm{ln}}^{ij}$ is to first compute the covariance matrix ...
2
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3answers
77 views

Probability distribution of $aX + b$, where a and b are integers and $X$ is a uniform variable

My textbook has the following introductory example about functions of random variables: Suppose that $X$ is a random continuous variable with a uniform distribution over the interval $(0,1)$. ...
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1answer
43 views

Relationship between random variables that are parameterized

Suppose we have $n$ random variables $X_n$ - let's say these are measures of customer engagement - and we sample these $m$ times through a set of designed trials. The resulting $m$ data points define ...
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1answer
50 views

What is the distribution of the maximum of two random integers chosen from [1,n]?

Suppose we have the set of integers from 1 to n. We proceed to choose two integers from this set at random and denote S = maximum of these two integers. What is the distribution of the random variable ...
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36 views

Appropriate Sample Size Formula

I have a uniform distribution set which contains different similarity scores. This similarity scores is linked with documents in which I am planning to examine. However, this set is big, and it is ...
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1answer
38 views

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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0answers
12 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 ...
2
votes
1answer
58 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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1answer
54 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
67 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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22 views

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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26 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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3answers
49 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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127 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] = ...
0
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1answer
43 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
58 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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0answers
20 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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18 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 ...
6
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1answer
137 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
56 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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0answers
49 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
52 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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0answers
26 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: ...
1
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1answer
32 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 ...
3
votes
1answer
71 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 ...
1
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1answer
42 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 ...
4
votes
2answers
79 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 ...
3
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0answers
47 views
1
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46 views

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
46 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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16 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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85 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
92 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 ...