Questions tagged [distributions]
A distribution is a mathematical description of probabilities or frequencies.
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Distribution of inverse Wishart to a power?
In a related question, I had asked about the norm induced by an inverse Wishart matrix. I am interested in generalizing that result somewhat. Let $A\sim\mathcal{W}_p\left(I,n\right)$, a Wishart matrix ...
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Are these equivalent (for p-values): super-uniform, stochastically larger than / dominating the uniform, conservative?
In the literature and online, I have found three different wordings that I think refer to the same concept: stochastically larger than uniform (which I take is ...
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How do I identify the "Long Tail" portion of my distribution?
I have a number of series that would typically be described as normal skewed or Gamma distributed. For example, in a group of customers I may have calculated their spend over a fixed length of time. ...
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Sum of absolute values of T random variables
Where X is a r.v. following a symmetric T distribution with 0 mean and tail parameter $\alpha$.
I am looking for the distribution of the n-summed variable $ \sum_{1 \leq i \leq n}|x_i|$.
$Y=|X|$ ...
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Finding the distribution of sample range for a Beta population
Let $X_1,X_2,\ldots,X_n$ be i.i.d random variables having density
$$f(x)=2(1-x)\mathbf1_{0<x<1}$$
I am trying to derive the distribution of the sample range $R=X_{(n)}-X_{(1)}$.
The usual way I ...
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Joint distribution of $Y$ and $S^2-Y^2$
Let $\{X_i\}_{i=1}^n\overset{iid}{\sim}\mathcal{N}(\mu,\sigma^2)$. Let
$\{b_i\}_{i=1}^n$ be a sequence of numbers so that $\sum_{i=1}^nb_i=0$
and $\sum_{i=1}^nb^2_i=1$. Define $$S^2=\sum_{i=1}^n(X_i-\...
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How to explain the difference between confidence and credible interval?
The key difference between Bayesian statistical inference and frequentist statistical methods concerns the nature of the unknown parameters that you are trying to estimate. In the frequentist ...
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What is the difference between a "population," a "sample space," an "underlying probability distribution? and a "model"?
I'm trying to understand an overview of the topic of statistical inference. I have learnt bits and pieces of many of the probability and statistics involved in it but before learning it rigorously it ...
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Strange connection between Bernouilli, Uniform and Geometric distributions
Final update on 11/29/2019: I have worked on this a bit more, and wrote an article summarizing all the main findings. You can read it here.
Let us consider $Z = X_1 + X_1 X_2 + X_1 X_2 X_3 +\cdots$ ...
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Square roots of sums absolute values of i.i.d. random variables with zero mean
In an earlier question, I asked about the limiting distribution of the square root of the absolute value of the sum of $n$ i.i.d. random variables each with finite non-zero mean $\mu$ and variance $\...
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What bounds can we place on approximation error for a moment-matching approximation with $N$ moments?
Suppose I have a distribution over the real line ($p$) and I'm approximating it by matching its first $N$ moments. What can I say about the approximation error as a function of $N$?
Alternatively, ...
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Goodness-of-fit for Discrete Distributions
I've been doing some data analysis with Scipy. So far I accomplished this with continuous distributions:
I can fit a probability distribution to a set of data points using a maximum likelihood fit. ...
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How to compare models with different distributional assumptions for response variable in GLM?
Let's say I have measurements $Y$ which are all positive, and the distribution seems to be somewhat skewed. I'm modelling $Y$ in GLM framework. Now I could set my GLM using different distributional ...
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Test for difference of distributions on a torus
I have two circular dependent variables and would like to test for a difference in the distributions (presumably circular means) between multiple treatment groups. There are a number of multivariate ...
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Exchangeable Processes over the Simplex
You are likely all familiar with Polya Urn process. I initially start with an urn containing $b$ black balls and $w$ white balls. At each step, I sample a black ball with probability $\frac{b}{b+w}$ ...
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Assign Numbers to N-sided Die to Make the Expectation Close to the Fair Mean
I got the following question in one interview: suppose we have an N-sided die and given the probability of landing on each side, how to assign values from 1 to N, to make the expected value close to $(...
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Distribution that doesn't belong to any maximum domain of attraction?
Question
Does there exist a (non-degenerate) distribution that does NOT belong to any maximum domain of attraction?
That is:
Does there exist any non-degenerate probability distribution function $F$ ...
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Goodness of fit to a fitted distribution
Assume we have a sample of data $\{x_1, \dots, x_n\}$ and a family of distributions $(f_\theta)_\theta$ indexed by some parameter vector $\theta$. We would like to fit $\theta$ to $\{x_1, \dots, x_n\}$...
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Distribution of the $L^2$ norm of a vector of components drawn from uniform distributions
We consider a random vector $\vec{v} = \left(x_{1}, x_{2}, \dots, x_{n}\right)$ built from $n$ real random variables drawn from a real continuous uniform distribution $\mathcal{U\left(a, b\right)}$, $...
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Concentration of maximum of subexponential random variables
I'm looking for a concentration bound on the maximum of a collection of sub-exponential random variables, which are not necessarily independent. More specifically, I have the following collection:
\...
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Covariance of order statistics convergence?
Suppose I have a sample $(X_1 \dots X_n)$ and $(Y_1 \dots Y_n)$, all of which are $N(0,1)$ random variables. I am interested in the asymptotic behaviour of
$$\frac{1}{n} \sum_{i=0}^n X_{(i)}Y_{(i)} $$...
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Expectation of a strictly increasing function
Assume that $X_1$ and $X_2$ are two i.i.d. random variables with pdf $f$. Also, assume that $a$ and $b$ are two fixed real numbers such that $a>b$. If $g$ is a strictly increasing function, do I ...
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Non-Uniform Spherical Distributions
Suppose $X_i\overset{\text{iid}}{\sim} N(0,1)$, and define the random vector $\mathbf{X}=(X_1,\ldots,X_n)$. Then the normalized vector $\mathbf{Z}:=\frac{\mathbf{X}}{\|\mathbf{X}\|_2}$ is uniformly ...
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sum of noncentral Chi random variables
if $X_1,...,X_n$ are independent random variables with noncentral chi distributions (same $df$ but different $\lambda$),
What is the distribution of $\sum_{i=1}^{n}{X_i}$
Just wondering if it can be ...
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What's the distribution of $|y-z|^2/|y-\bar{y}|^2$ for vectors with i.i.d. standard normal coordinates?
Let $y_1, y_2, \ldots, y_n$ and $z_1, z_2, \ldots, z_n$ be samples of size $n$ of a normal distribution $\mathcal{N}(0,1)$. My goal is to find the distribution of
$$\frac{\sum_{i=1}^n (y_i - z_i)^2}{\...
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Photo attractiveness rating experiment: significance and effect size
I'm a photographer working on a personal project where I experimentally test dating photo advice. I take photos of people under different conditions, changing one variable while keeping others ...
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Finding the probability density function of Hotelling's T-squared distribution
The following image is seen on wikipedia when searching for Hotelling's T-squared distribution
This is apparently the pdf of the Hotelling T-squared distribution at different parameters. However, I ...
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Distribution $f$ that minimizes $JSD(f||q) + JSD(f||p)$
What can we say about the distribution $f^*$ that is the solution to the following optimization problem:
$$\min_f JSD(f||p)+JSD(f||q) ,$$
where $p,q$ are given distributions over some set, and $JSD$ ...
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Integrating the inverse-Wishart density
It is alleged in this question and in the Wikipedia article and elsewhere that the density function for the inverse-Wishart distribution with $n$ degrees of freedom on $p\times p$ positive-definite ...
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Is there an asymptotic distribution expansion that does not go negative when truncated?
I am looking into a problem where I have a distribution that converges to the normal distribution as its parameters become large. I am looking at the rate of this convergence, and seeking to describe ...
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Kolmogorov distributuon derivation
I would like to know if there is a book talking about the derivation of Kolmogorov distribution (Using usual definition for the bridge process)
\begin{align}
P(K\leq x)=1-2\sum_{i=1}^{\infty}(-1)^{i-1}...
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How were statistical distributions discovered?
Let me start, that i know that it's not very difficult to generate a probability distribution. If one takes any positive integrable function and normalizes it, this results in a probability density. ...
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Correcting Sample Selection Bias given actual Distribution
I have two datasets, both from the same population:
The samples from the first survey are quite representative of the underlying truth. However, the second survey comes with a change in distribution ...
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Distribution of conditional expectation?
Let $X,Y$ be random variables with pdf $f_{X,Y}$. I would like to find the distribution of the random variable $\mathbb{E}(Y\mid X)$, conditional expectation of $Y$ given $X$. If a specific form of $\...
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Motivation behind the definition of a heavy-tailed distributions
The current Wikipedia definition is
The distribution of a random variable $X$ with distribution function $F$ is said to have a heavy (right) tail if the moment generating function of $F,$ $MF(t),$ ...
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Generalization of Dirichlet distribution over matrix
I know there is generalization of normal distribution of matrix-valued random variable, i.e., Matrix normal distribution. I wonder whether there is generalization of Dirichlet distribution that each ...
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Exact Null Distribution with Ties
I am interested in deriving exact null distributions for small-sample test statistics with non-trivial ties. Not fundamentally continuous variables that happen to have a few repeated values, but ...
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Processes behind statistical distribution laws: a compendium?
The simple processes that "explain" the binomial, Gaussian or Poisson distribution are relatively well-known. Johnson or shot noises may be known in restricted area of science. Sometimes, a ...
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How can I calculate the probability that the product of two independent random variables does not exceed $L$?
I have one variable, $X$, which is provided hourly for a period of one month (720 total values in the series). I have another variable, $Y$, which is provided quarterly (for which I am provided the ...
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Sampling distribution of the skewness / kurtosis from non normal distributions?
Is there some sort of approximation or analytic definition of the sampling distributions of the skewness and kurtosis when samples are taken from a NON-normal distribution?
I have been looking for ...
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Use Chi-Squared or Binomial Test if Distribution is not Known?
Suppose you have a set of data (eg. [a, b, a, a, b, b, etc.]), and you have the suspicion that the set of data follows the binomial distribution.
Your Null Hypothesis is: The probability of success ...
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Asymptotic Distribution of the Wald Test Statistic
I am trying to understand the asymptotic distribution of the Wald test statistic, specifically under the alternative hypothesis which I've found little reference to.
For clarity, the binary ...
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Probability distribution over classes as labels in classification task
Classical classification problem has next formulation.
Given a set of $n$ attributes, a set of $k$ classes and a set of labelled training instances:
$(i_i, l_j),...,(i_j, l_j)$, where
$ i = (v_1, v_2,...
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I'm not asking for a conjugate prior. Is there a distribution $p(x|y)$ that satisfies $\int p(x|y)Beta(y|a,b) dy = Beta(x| a', b')$?
I know the result of integrating a Gaussian against another Gaussian is still Gaussian, $$\int N(x|\mu_y,\sigma_y)N(y|\mu,\sigma) dy = N(x|\mu',\sigma')\quad.$$
Can I get the same form for Beta ...
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Dispersion parameters in GLM
I'm trying to find the motivation behind the extended form of the exponential family of distributions in the fundamental paper on GLM by Nelder and Wedderburn (Generalized Linear Models, J. R. Statist....
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Testing a proportion in an online setting
I work in an online security setting. My goal is to detect if the number of locked accounts per time unit is stable or not. I've tried several approaches, detailed below, but I am not satisfied yet. ...
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Uniform Convergence of Moment under Empirical distribution
Let $X$ be standard Gaussian random variable with cdf $F(x)$. Let $\{X_i\}_{i=1}^n$ be a sequence of i.i.d. standard Gaussian random variables. And let $F_n(x)=\frac{1}{n}\sum_{i=1}^n1_{\{X_i\leq x\}}$...
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Distribution of ratio of Poisson distributed random variables
I am currently reading a paper and puzzling about a certain statement. I have a ratio
$\frac{\hat\alpha_{T+1}}{\hat\alpha_{T}}=\frac{H_{T-4}+...+H_{T}}{H_{T-3}+...+H_{T+1}}\left(\frac{D_{T-3}+...+...
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Theoretical justification of choice for confidence interval exact method for the success probability parameter of negative binomial variable?
I have a computer experiment that runs the Bernoulli series with unknown probability $p$ of success. The experiment terminates when $m$ failures are observed. So, the unknown parameter $p$ has the ...
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Haar prior for von Mises distribution
Ok, Let me tell you that this is the very first time that I have no idea with the question below. I can not find a solution or anything that will lead me to it. I say this to prevent comments "what ...
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