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).
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Does $f(X) = X$ imply $f$ is identity?
For a random variable $X$, does $f(X) = X$ imply that $f(.)$ is the identity function, at least over the support of $X$?
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Joint distribution of X and Y given both of X and Y are random variables from normal distribution, but X and Y are correlated with ${\rho}$ [closed]
The difference between being two independent random variables and being uncorrelated?
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Can VAE treat each pixel rather than image as random variable? [closed]
As I understand, the VAE encoder outputs vector of means and variances with each element corresponding to one image in the X training set. This means that we treat each image as one random variable . ...
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Empirical Histogram and PMF
I am taking an introductory course on statistics/probability and there's a concept that I am confused with.
That is the difference between empirical histogram vs PMF.
First off, let's use the example ...
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Binomial glmm with a categorical variable including zero cell [duplicate]
I am analyzing the data using generalized linear mixed-effect models (GLMM) using the lme4 software package (Version 1.1-30 in R; Bates, Maechler, Bolker, & Walker, 2022).
The model includes 2 ...
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Weird definition of negative binomial distribution
In a paper I am reading, they define the negative binomial as the following: random variable $X$ has a negative binomial distribution with parameters $p \in (0,1),k \in \mathbb{N}$ if
$$\mathbb{P}[X=t]...
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The PDF of the random variable Z=Y+Y*X
I have two independent random variables, $Y$ and $X,$ where
$Y$ is a random variable with a Gaussian distribution and a zero mean.
$X$ is a random variable with a Gaussian distribution and a zero ...
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Sum of i.i.d. random variables for which Chebyshev inequalities are tight
Chebyshev's inequalities: Let $X$ be a random variable with finite expected value $\mu$ and finite non-zero variance $\sigma^{2}$. Then for any real number $\delta > 0$,
$$ \Pr[|X - \mu| \geq \...
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Possible to have independent observations without random sampling?
Is some form of random sampling required for observations to meet the "independence" assumption of various types of regression analysis and hypothesis testing?
My data (fish counts, without ...
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Analyzing data from a non-randomized sampling design (ecological monitoring)
I have 2 questions about analyzing data that was not randomly sampled from a population.
I work with "ecological monitoring" data that involves repeatedly taking measurements from the same ...
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Choosing a proposal density g(x) for $ f(x)= {\Large \frac{e^{x}}{(e-1)} }$ [closed]
In finding an proposal distribution function $g(x)$ for the following function:
$ f(x)= {\Large \frac{e^{x}}{(e-1)} }$ where $0 \leq x \leq 1$
Tested with $$x^2+1, 1/x+1$$ and other variations, but ...
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Confused about random variables and time series
I have this two practical (with R) exercises for my exam:
Using $\chi^{2}$ to test (goodness-of-fit) if fluctuations in wind speed are Gaussian.
Calculating Lagrangian time scale for wind speed. In ...
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What are the names of random variables in conditional probabilities?
In a conditional probability like p(A|B), what A and B are called?
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Problem with probabilities of functions of continuous random vector
I have problems with a true or false exercise: if the joint pdf of the continuous random vector $(X,Y)$ is
$$f(x,y)=
\begin{cases}2x,& 0 \le x \le 1, 0 \le y \le 1,\\
0,& \text{otherwise}
\end{...
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Why is E[X+Y|X,Y]=X+Y?
Intuitively, it seems obvious, but I am struggling to prove it for the case where $X_1, ..., X_n$ are continuous random variables. I am aware that $E[c(X)|X]=c(X)$. So how would one show that $E[c(X_i)...
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Covariance matrix of multivariate normal when negative values are made zero
Let $x$ be $n$ dimensionally multivariate normally distributed with mean $\mu$ and covariance matrix $\Sigma$. Now let $y$ be random variables defined by
\begin{equation}
y_i= \begin{cases}
0, ...
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How should we interpret random vectors in the context of statistical inference?
I am reading the book "Mathematical Statistics" from Jun Shao and I got some questions.
Here is the part of the book that I am struggling with
"In statistical inference and decision ...
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Complicated Transform of a Beta Random Variable
Consider a Beta distributed random variable, $X \sim Beta(a,b)$ Then consider the transform $$Y = \sqrt{K\frac{X}{1-X}}$$ where $K > 0$ is a constant.
How could you go about finding the PDF of $Y$?...
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Distribution of the Square Root of a Beta Prime Random Variable
Given a Beta Prime distributed random variable $X \sim BP(a,b) $ with probability density $$\rho_X(x) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\frac{x^{\alpha - 1}}{(1+x)^{a+b}}, x > 0$$ Consider ...
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What is the expected value of $X_i/\|X\|^2$ when $X \sim \mathcal{N}(\mu, \sigma^2I)$
Let $X$ be an N-dimensional normal random vector with non-zero mean $\mu$ and diagonal covariance matrix $\sigma^2I$. I would like to understand if it is possible to derive the expected value of the ...
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Probability of multiplication of independent random variables
Let X,Y,Z be three independent continuous random variables with CDFs $F_X,F_Y,F_Z$, respectively. How does one find $P(c x\leq yz)$ where c is some constant? is the following correct?
\begin{align}
P(...
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Is the variance of an estimator a random-variable? [closed]
Is the variance of an estimator a random-variable?
If so, the mean of the variance and the variance of the variance exist.
An estimator of the variance of this variance of an estimator also exists.
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Square of a zero random variable
Following up on this question and the answer:
Bootstrap variance of squared sample mean
Summary: $X_1$, ..., $X_n$ are IID. Define
$$
\overline{X}_n=\frac{1}{n}\sum X_i
$$
and $T_n=\overline{X}_n^2$. ...
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How to condition on a minimum value to make an inflated distribution
I was watching a video on generating forecasts for lead time. In this video there was three charts.
Lead time observations.
The "smooth" distribution is a mixture of Poissons where the ...
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Random walk and Poisson process
(1) A point is chosen at random in a circle with center at the origin and radius R. That point is taken as the center of a circle with radius X where X is a random variable having density f. Find the ...
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What is the distribution of the sum of differences of i.i.d. variables?
Let's suppose that $X, X_1, X_2, \ldots, X_n$ are i.i.d. continuous random variables.
Let's define $D_i$ as the difference between $X_i$ and $X$:
$$
D_i = X_i - X
$$
My end goal is to derive the ...
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Is the sample proportion ($\hat p$) a random variable?
Since it should vary from sample to sample, I suppose it should be a random variable. But if it is, when we write the variance of sample proportions, should we write an uppercase P-hat as the index ...
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Creating auto-correlated random time series
I want to create random time series that follow a given auto-correlation function. For this I am using an AR(n) model approach:
$$X_t = \sum_{i=1}^n\alpha_i X_{t-i} + \epsilon_t$$
where $\epsilon$ is ...
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Upper bound of P[X < Y]
X and Y are independently distributed discrete random variables. is it possible to find an upper bound for P[X<Y] that is always less than or equals to 1?
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Calculation of conditional variance for 2 correlated random variables
If we have a bivariate normal distribution for $\left(X, Y\right)$ then, we can calculate the conditional mean and variance of $\left( X|Y \right)$, as demonstrated here https://online.stat.psu.edu/...
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Random selection with higher probability of selecting part of the range?
I'm trying to write an algorithm/formula to select a random number between 0 and 1 but I want there to be a higher probability of selecting a number towards the top of the range (or bottom in certain ...
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What is the conditional probability $P(A|B)$ in measure theory?
In Schervish's Theory of Statistics (1995) and again in A Measure Theoretic Formulation of Bayes' Theorem by @ArtemMavrin, the following equation was proved in detail
$$
\mu_{\Theta \mid X}(A \...
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Does k moments imply k + $\epsilon$ moments?
If you have a random variable $X$ such that $\mathbb{E}(|X|^k) < \infty$, does it follow that $\mathbb{E}(|X|^{k+\epsilon}) < \infty$ for some (potentially small) $\epsilon > 0$? If not, ...
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What does a random sample mean in linear regression? [duplicate]
For linear regression
$$y_i=x_i^T \beta+\epsilon_i$$
what is been sampled? Sometimes I see that our sample is $\{X_1,X_2,\ldots,X_n\}$ while other times I see that our sample is $\{Y_1,Y_2,\ldots,Y_n\}...
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Variance of a linear combination if parameters are random instead of fixed (as in bayesian estimation vs MLE)
If X is a r.v. and beta a fixed parameter (say, an MLE estimate), we know that: Var(beta*X) = beta^2 * Var(X).
I wonder whether this holds in a bayesian framework, where beta is itself a r.v.
Thank ...
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How do we choose the sigma algebra for a specific experiment?
I am trying to understand the connect between probability and measure theory. I get some basic things about measure theory. But what I still don't get is how do we choose the sigma algebra for our ...
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Can CDF of a real random variable be a complex function? What does it mean physically?
I have a random variable $X$ which follows the following probability density function,
$$ p(x) = \frac{1}{4\pi} \Big[ \operatorname{erf}\Big(\frac{k\mu-x+2\pi}{\sqrt{2}k\sigma}\Big) - \operatorname{...
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Example of two dependent RVs where all subsets of first and second RV has no correlation?
From Simple examples of uncorrelated but not independent 𝑋 and 𝑌, we have:
𝑋 ∼ 𝑈(−1,1)
<...
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Mutual Information between the mean of normally-distributed random variables and another variable
Given normally-distributed (possibly dependent) random variables $X_1 \dots X_n$, their mean $X=\frac{1}{n} \sum_iX_i$, and another discrete r.v. $Y$, can we relate $MI(X,Y)$ with the individual $MI(...
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Variance of a symmetric random variable
Let say I have a multivariate random variable $x$ with dimension $s$ which is symmetric at 0 and not necessarily be a multivariate normal distribution. Then what is the variance of $Ax$, where $A$ is ...
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powerSim will not run and repeatedly gives me the warning: boundary (singular) fit: see help('isSingular')
I am attempting to perform a power test to determine the minimum sample size of my next upcoming experiment. First, I made a data set:
...
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Is the score-function (gradient of likelihood function) a random-variable?
Wikipedia:The score is the gradient (the vector of partial derivatives) of $\log \mathcal{L}(\theta)$, the natural logarithm of the likelihood function, with respect to an $m$-dimensional parameter ...
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Probability of $A<B$ when $A$, $B$ are random variable with different distribution?
Preparing exams, I ran into the following problem:
Edit: it shouldn't be represented as it was. Added the storkes.
Let $A$, $B$ be two independent variables having probability distribution:
$$
\...
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Moment generating function of a constant multiple of a random varaible
Let $Y$ be a random variable which is a function of another random variable $X$, such that $Y=aX$, where $a$ is a constant. Is it possible that the moment generating function (MGF) of $Y$, is given by,...
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What is the sum $\sum_{m} e^{i (U_m k + \beta_m)} $ when $U$ and $\beta$ follow different distributions
I have the following function.
$$ x(k) = \sum_{m} e^{i (U_m k + \beta_m)} $$
$i = \sqrt{-1}$
Here, $U_m$ are samples drawn from a Gaussian random distribution.
$$ U_m \sim \mathcal{N}(\mu, \sigma) $$ ...
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Means to estimate the decay rate of a noisy decay process?
I have a decay process which appears essentially like
$$ f(t) = \xi(t)\exp[-t/\tau],$$
where $\xi(t)$ is a stationary Gaussian noise with some mean, variance, and correlation function.
Given a ...
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What does it mean for a pair of data points to be i.i.d?
I am studying conformal prediction where they mention that the data points in the calibration data should be i.i.d.
Here's what it reads -
we reserve a moderate number
(e.g., 500) of fresh i.i.d. ...
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If X $d=$ Y, Y $d=$ Z, then X $d=$ Z?
Let X, Y, and Z be 3 independent random variables. If X has the same distribution with Y and Y has the same distribution with Z, will X has the same distribution with Z? I think it is right but don't ...
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how to view conditioning on a random variable rather than a particular value of that variable
The question is about $P(Y|X)$ versus $P(Y|X=x)$. Is there an alternate way to write $P(Y|X)$
that makes its meaning more clear?
I believe these are correct equations for conditional entropy:
$$
H(Y|...
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A question about product and division of random variables
Taken three real continuous random variables $X,Y,Z$, non negative (to simplify the handling of the problem),
with respective pdf's $p_X , p_Y , p_Z$ .
Then if
$$
Z = XY\quad \left| {X,Y\;indep.} \...
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