Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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39 views

Why Pearson correlation coefficient follows beta distribution? [duplicate]

I understand that the pearson correlation coefficient r can be tested using student-t distribution. Also I can understand the exact distribution of r from https://en.wikipedia.org/wiki/...
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266 views

Fitting measured, real-world data to theoretical distribution: How to test goodness?

I have a large sets of real-world user data (30k, 80k, 90k measurements). To be precise those are simply session lengths for a specific system. I want to create a theoretical model of this, to ...
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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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Maximum Likelihood vs. Restricted Maximum Likelihood vs. Penalized Maximum Likelihood

I am trying to better understand the differences between the Maximum Likelihood vs. Restricted Maximum Likelihood vs. Penalized Maximum Likelihood. In general, here is what I understand so far: The ...
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3answers
3k views

What is "the shortest half of the data"?

Here is a histogram (realized with JMP) displaying two types of box plot called outlier box plot and quantile box plot. Right below, there are a bunch of explanations of the meaning of the different ...
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1answer
165 views

When you do a random permutation F test (by permuting group membership) is inference made on the samples or the populations?

I am trying to understand different approaches that use randomization procedures. One thing I cannot come to a clear conclusion on is this: Say we have randomly sampled individuals from 2 populations. ...
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8 views

Extreme values and skewness in data for machine learning

I have 2 classes that I want to train a model with. Looking at histograms of the features shows that most of the features have some very large outliers, and some of the distributions are skewed. My ...
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28 views

A mixture of discrete and continuous components

Suppose that random variable $X$ is sampled from Bernoulli with probability $\pi$. Let $Y\sim N(\mu, \sigma^2)$ and denote $Z=XY$. Then, when $X=0$, $Z$ equals 0 (and this happens with probability $1-\...
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What are the standard statistical tests to see if data follows exponential or normal distributions?

What are the standard statistical tests to see if data follows exponential or normal distributions?
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83 views

Biased urn experiment and Fisher's noncentral hypergeometric distribution

Let's have an urn with $m_i= 2$ balls for $c=3$ different colours with different weights $w_i$. $n=2$ balls are taken randomly but the probability of sampling a particular coloured ball is ...
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262 views

How do I fit scale and location parameters of the Student's t distribution?

I would like advice on how to correctly set the parameters (loc and scale) for the student T distribution that best fits my data of daily stock returns. I'm pulling random numbers from a student T ...
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Do all observations arise from probability distributions?

Below is the quote from Karl Pearson in the book: “The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century” by David Salsburg: Over a hundred years ago, Karl Pearson ...
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191 views

Sample without replacement from 1 to N and stop when the value is less than the previous one

I came across a question as below Assume a series includes integer from 1 to N. Every time one samples an integer without replacement from the series. The process continues if $X_{n}$ >= $X_{n-1}$, ...
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Can two subsamples of the same dataset have different distributions (covariate shift)?

The reason for my question is that I trained a model for binary classification; once obtained the results, I trained another model on these results where: The predicted instances are used as a ...
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1answer
27 views

Deciding the Number of Clusters : Standard Methods vs. Non-Parametric Methods

I was watching this video over here (https://www.youtube.com/watch?v=UBiaLq5V7mE) that discussed a Non-Parametric based Bayesian approach for deciding the number of clusters in a dataset. Essentially, ...
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29 views

Does any Probability Distribution Functions Exist that can Model this Problem?

I invented these two problems which can might correspond to a probability distribution - but I am not sure which ones. Case 1: Suppose there are 5 "hats" : the first "hat" has 10 ...
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169 views

Fitting a distribution to my timeseries: two R-packages, two contrary results

typical R-User here, applying a bunch of packages to my data, hoping for a convincing result although I understand only half of what I do at most (and then getting rejected by the reviewers, surprise)....
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Can chi-squared distribution be left-skewed?

I'm learning chi-squared distribution and know that as degree of freedom increases, right-skewed chi-squared distribution will approximate to normal distribution shape. I'm wondering can its ...
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Maximum Uncertainty in Normal Distribution

While reading Goodfellow's Deep Learning Book, I came across the below fact about Normal Distribution. I am not sure I have understood what led to this conclusion, Can someone help with it? "Out ...
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1answer
108 views

The distribution of the largest index in a sample that is smaller than the median of the population

Consider a sorted sample $\vec{x}(n)$ with size $n > 100$ ($n$ is odd) from some rv $X$. Let $m$ denote the median of $X$. Let $R(\vec{x}(n))$ be the largest index $i \in [n]$ s.t. $x_i \leq m$. ...
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When $(X_1-X_0, X_1-X_2)\sim (X_2-X_0, X_2-X_1)\sim(X_0-X_1, X_0-X_2)$?

Consider a bivariate distribution function $P: \mathbb{R}^2\rightarrow [0,1]$. I have the following question: Are there necessary and sufficient conditions on $P$ (or on its marginals) ensuring that $$...
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Is this the Equivalent of "Data Snooping" in Bayesian Models?

I have the following question about "Data Snooping" in Bayesian Models. In Bayesian Models, priors are generally said to come from an independent source, such as domain experts on the data ...
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Is a uniformly random number over the real line a valid distribution?

Is $\text{Unif}(- \infty, \infty ) $ a valid distribution? I'm trying to capture the idea of a completely random number (where every number has an equal chance of being chosen), but I'm not sure if ...
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How to break up the following scale into equal segments?

In Excel, there is a conditional formatting feature like this: Let's say in order to figure out what color to use for a value, I split the gradient into 14 steps, such as: Red:6Parts,Yellow:0Parts ...
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194 views

Find $P\{ (A \, \text{or} \, B) \, \text{and} \, (A_1 \, \text{or} \, B_1) \}$ or a lower bound in this specific case

Define $X$, $Y$, $X_1$, $Y_1$, and $Z$ to be some positive random variables, for each of which we know the distribution. Note that these variables are independent of each other. Let $t, a$ two ...
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23 views

Density of cost-per-loss random variable

Given a random variable $X$ with support $(a,b)$ and a value $d\in (a,b)$ (the "deductible"), the cost-per-loss random variable associated to $X$ is given by $$Y=\begin{cases}0&\text{if }...
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107 views

Find $\mathbb{E}\{M W V V^H W^H M^H \}$ in the following case:

Let $M$ be an $n \times p$ matrix with complex Gaussian elements with mean $= \mu$ and variance $= \sigma^2$. Also, define matrix $W$ as a $p \times m$ matrix for which the columns are of unit norm. ...
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1answer
34 views

How can I maximize statistical power for small sample sets with a known, potentially non-normal, distribution?

I want to estimate the mean difference of a variable between populations with some level of confidence and with a minimum number of samples collected. The populations share the same variance and ...
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1answer
21 views

Statistical analysis of distributed values in Java [closed]

I am writing a program in Java that outputs a List<Double> of distances that roughly follow a bell curve distribution. From this data, I need to generate two ...
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317k views

When (and why) should you take the log of a distribution (of numbers)?

Say I have some historical data e.g., past stock prices, airline ticket price fluctuations, past financial data of the company... Now someone (or some formula) comes along and says "let's take/use ...
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48 views

Are these distributions quantiles different? statistical significance

I have an experimental distribution and I am looking at the quantiles. For example the counts for the quantiles are [10, 20, 40, 10, 20] respectively and I want to know if they differs from a uniform ...
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1answer
50 views

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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23 views

How are these distributions same?

I am reading this paper 'Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations' and struggling with understanding the proof of their main claim (in Appendix A). ...
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2answers
46 views

The product of two lognormal distributions may not be a lognormal

If two lognormal random variables $X, \:Y$ are neither independent nor jointly normally distributed, the product $XY$ may not be lognormally distributed. Where can I find the explanation for the 'may ...
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1answer
481 views

Visually compare binomial and beta-binomial appropriateness in R

I'm trying to choose the best distribution family for generalized linear regression. My outcome is cross-sectional, over-dispersed proportion data (# of behaviors/20-22 possible behaviors). I used the ...
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1answer
46 views

What is "natural" about the natural parameterization of an exponential family and the natural parameter space?

In Section 3.4 of Casella and Berger's Statistical Inference, an exponential family is defined to be a set of pdfs or pmfs such that for each member $f(x | \boldsymbol{\theta})$ of the set, $$ f(x | \...
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28 views

Ratio of two independent Gamma distributions

Pre-requisites I have seen in Wikipedia that it says for independent r.v. $X \sim G(m), Y \sim G(n)$ the ratio $X/Y \sim B'(m,n)$. Where $G(x)$ denotes the standard Gamma distribution $$ f(x,\lambda)=...
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2answers
58 views

How to model a probability distribution to phone call duration?

I was plotting the phone call durations, and the result seems like a right-skewed histogram. Shape-wise, Poisson, Raylegh, Chi-2 can be candidates for modeling that. But is there a more rigorous way ...
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1answer
19 views

Simulation of a truncated normal distribution over two intervals

Given $X$ a random variable with a normal distribution, what is the best procedure to simulate $X|X\in[a;b]\cup[c;d]$, i.e. we want to simulate the truncated normal distribution only on the intervals $...
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1answer
15 views

What is the appropriate distribution of Ordinal Data?

I have an ordinal random variable {0,1,2,3}. The variable should describe disease severity. What kind of probability distribution is the best fit to model this variable? The categorical distribution ...
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1answer
319 views

Distribution of $x_4(x_1-x_3)+x_5(x_2-x_1)$ with iid $x_i \sim N(0,1)$

Problem Given are 5 independent standard normal variables $x_1,x_2,x_3,x_4,x_5$. What is the pdf of $x_4(x_1-x_3)+x_5(x_2-x_1)$ ? What I know $$x_1-x_3\sim \mathcal{N}\left(0,\sqrt{2}\right)\tag{1}$$ $...
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1answer
54 views

Difference distribution of iid noncentral chi2 variables

Given are two identically distributed independent noncentral chi-square variables $$Z_1\sim \chi^2(k,\lambda)\\Z_2\sim\chi^2(k,\lambda)$$ with $k=2$ degrees of freedom and location parameter $\lambda$....
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1answer
222 views

convolution and deconvolution of random variables of different dimensions

Preliminary: Let's say we have $Y=X+Z$ ($Y$ is data, $X$ is latent variable and $Z$ is noise), where the random variables are all in $\mathbb{R}$. Then an inverse Fourier transform leads to \begin{...
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Why are Ratios "Dangerous" in Statistical Modelling? [closed]

Why are Ratios "Dangerous" in Statistical Modelling? A friend was telling me today that it is unwise to use the ratio of two variables as a variable in a regression model, and that is better ...
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1answer
585 views

Canonical form and exponential family

Suppose you have a random variable X, who's distribution depends on $\theta$. If X is a part of the exponential family of distributions, X can be written in a certain form, namely: $$f_\theta(x)=h(x)*...
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1answer
47 views

How are copulas used in the real world?

I have been reading about copula models. Essentially, copula models seem to be a creative method for creating a joint probability distribution from several variables, in which each individual variable ...
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1answer
1k views

covariance matrix of residuals from a fitted model to decorrelate residuals

I fit a geeglm model with clustered data and now I would like to decorrelate the residuals of the model in order to run model diagnostics. I read that if I can obtain the covariance matrix of the ...
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Cramér–von Mises criterion vs Goodness of Fit

I had the following question about the Cramér–von Mises (CVM) criterion vs Goodness of Fit As described above, it seems like the CVM test measures how close the empirical probability distribution ...
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28 views

How to deal with bimodal feature in Logistic Regression?

I am working on a binary classification problem where one of the most interesting features has a distribution which looks more or less bimodal. Here is the distribution plot of that feature: The two ...
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2answers
435 views

Should I use the Log-Likelihood (G2) test to compare word frequency across corpora?

Apologies for cross posting this I posted in Linguistics but so far there have been no takers. I need to compare the frequency of occurrence of a word in two corpora. Take for example the word fail ...