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4
votes
1answer
69 views

Distribution Reference $\gamma x^{\gamma-1}$

I have been unable to find resources regarding the family of distributions with pdf $$ f_\gamma(x) = \begin{cases} \gamma x^{\gamma-1}, & \text{if } 0 \leq x \leq 1 \\ 0, & \text{...
0
votes
0answers
24 views

Generalizing Bayesian methods by assuming a “distribution of distributions” instead of a prior

Bayesian methods assume a prior distribution with several hyperparameters. Unfortunately, this is asymptotically incorrect, because distributions in the real world are never exact. For example, the ...
0
votes
0answers
18 views

Dimension of a probability distribution function

Consider the following statement We want to sample from a complex high dimensional distribution which is intractable. What is meant by a dimension of distribution here? Does each random variable ...
0
votes
0answers
10 views

Wilks' lambda's exact distribution when one of the parameters is 1 or 2

Citing Wikipedia, From the relations between a beta and an F-distribution, Wilks' lambda can be related to the F-distribution when one of the parameters of the Wilks lambda distribution is either 1 ...
0
votes
0answers
17 views

Standard Notation for Gaussian Measure?

There are standard notation for normal PDFs and CDFs, being $\varphi$ and $\Phi$, respectively. I would like to know if there is also standard notation for the Gaussian measure corresponding to a ...
3
votes
1answer
110 views

What can we say about distributions of random variables $X$ such that $X$ and its inverse $1/X$ have the same distribution?

What can we say about random variables such that it and its inverse have the same distribution? One example is Cauchy distributed random variables, easily proved via the fact that if $X, Y$ are IID ...
1
vote
1answer
82 views

Most powerful test for deciding probability mass function

Let $X$ be an integer valued random variable supported on $\{0.1.2...,12\}$ whose pmf is either $g(x)=1/13; x=0,1,...,12$ or $ f(x)=\dfrac x {36} 1_{\{0,1,...,6\}} + (\dfrac 13 - \dfrac x{36})1_{\{7,...
3
votes
1answer
66 views

How do I learn when to apply which statistical distributions? [closed]

I am learning Bayesian modeling and having trouble keeping straight when it is most appropriate to apply the various statistical distributions beyond the basic ones like the beta, binomial, and ...
3
votes
0answers
204 views

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. ...
1
vote
0answers
18 views

Tools for self-study: Constructing and understanding systems of distributional families

I am looking for a resource that probably does not exist, but, well, hope springs eternal. I have become increasingly interested in the process by which distributions are discovered or invented. ...
2
votes
0answers
56 views

rule for Normality skewness $<|2.0|$ , kurtosis $<|9.0|$?

Quoting: As can be seen in Table 1, the experimental and control group distribution were sufficiently normal for the purpose of conducting a $t$-test (skewness $<|2.0|$ , kurtosis $<|9.0|...
3
votes
0answers
415 views

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: \...
0
votes
1answer
30 views

Random variables stable by nonlinear function

Let $h$ a function and $X$ a random variable with CDF $F$. We say that $X$ is stable by $h$ if $h(X)$ follows $F$. I would like to know if there is a literature for those kind of random variables? (...
1
vote
0answers
27 views

Pedagogical derivation of different PDFs

When I see the equations of a t distribution, or a Gamma, or a Chi squared I feel sometimes puzzled and ask to myself: how did they arrive to such not immediate intuitive equation? I've seen ...
0
votes
1answer
46 views

Book recommendation for understanding more about distributions in econometrics/data analysis

I have taken a few econometrics courses and I know some things about distributions; things like skewedness, kurtosis, ect. However, there are other areas regarding distributions that I know I am weak ...
16
votes
6answers
546 views

How could I have discovered the normal distribution?

What was the first derivation of the normal distribution, can you reproduce that derivation and also explain it within its historical context? I mean, if humanity forgot about the normal ...
2
votes
0answers
61 views

Tukey-Lambda Distribution versus Student's t Distribution

On Wikipedia and many other sites, it is mentioned that The Tukey lambda distribution is actually a family of distributions that can approximate a number of common distributions. For example, ...
5
votes
3answers
3k views

Implementing a discrete analogue to Gaussian function [closed]

Given a Gaussian function of the form $$g(x) = ae^{-(x-b)^2/(2c^2)}$$ I am interested in a discrete analogue to this, which deals with the case where $x$ is discrete. As I understand there are two ...
1
vote
0answers
39 views

Are there any statistical model that could learn an arbitrary probability distribution?

I have the follow problem: I need a statistical model or machine learning method that allow me to parameterize a probability distribution into a finite number of parameters $\theta$. That's mean, ...
2
votes
0answers
56 views

Good Text for Learning about Multivariate Distributions [duplicate]

I have just started learning about multivariate distributions using Mathematical Statistics with Applications by Wackerly, Mendenhall and Scheaffer but I find their explanation a bit confusing, could ...
3
votes
1answer
58 views

Understanding how to apply goodness of fit tests when parameters of a continuous non-normal distribution have been fitted to the data

I have some data and wish to fit several distributions to it, many of which are compound and/or complex. I'd like to know whether a given family of distributions is appropriate which I can apparently ...
16
votes
2answers
300 views

Why is there -1 in beta distribution density function?

Beta distribution appears under two parametrizations (or here) $$ f(x) \propto x^{\alpha} (1-x)^{\beta} \tag{1} $$ or the one that seems to be used more commonly $$ f(x) \propto x^{\alpha-1} (1-x)^{...
2
votes
1answer
41 views

Reference request for Boltzmann's Theorem on maximum entropy distributions

Wikipedia (https://en.wikipedia.org/wiki/Maximum_entropy_probability_distribution#Distributions_with_measured_constants) describes in detail a theorem which gives the maximum entropy probability ...
0
votes
1answer
40 views

An article that explains what happens to CNNs when the training and testing data follow different distribution

Is there any good paper (sort of best practices) that experimentally explains what happens to CNNs when the training and testing data follow different distribution. This is for the case when CNN is ...
1
vote
1answer
84 views

Comparing estimation procedures via Monte Carlo simulations

I am looking for references of studies that show how to use Monte Carlo simulations to compare different estimators for any given parameter of any probability distribution (for example, comparing the ...
2
votes
0answers
122 views

Generalization of the Irwin-Hall distribution for general linear combinations of uniform variables?

Consider the random variable $Z$, defined by: $$Z = \sum_{k=1}^n c_k X_k$$ where $X_k \sim U[0,1]$ is a real random variable with continuous uniform distribution between 0 and 1, and the $c_k$ are ...
1
vote
1answer
72 views

Why does the author subtract the minimum $\ln\,\ln([1-F(t)]^{-1}])$ for the Weibull plot?

When I read the Weibull Analysis Handbook (Abernethy et al., (1)), in Table 2.1 (p15), I don't really know how he gets the right-hand column ("Col 2 value - Min Col 2 value (-6,91)). Does he assume it ...
4
votes
0answers
56 views

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 ...
8
votes
2answers
475 views

Where can I get information about relationships among probability distributions in statistics?

I'm interested in relationships among distributions. Like 'Sum of exponential random variables is a gamma random variable. Certain conditional distribution is another distribution etc.' I searched ...
4
votes
2answers
954 views

Distribution cheat sheet for Bayes data analysis

Has anyone developed a "cheat sheet" of sorts that describes the appropriate use of distribution types for different types of data? For example, beta for coin-type data (e.g. Therapy versus control), ...
0
votes
0answers
43 views

Estimating the Parameters of Multivariate Gaussian from Conditioned Distributions

My goal is estimating the distribution parameters of a multivariate Gaussian $\mathcal{N}(\mu,\Sigma)$ in $\mathbb{R}^n$ from observations that were generated from different conditioned variants of ...
5
votes
2answers
532 views

Variance of the modulus of a random variable

Let $X$ be a random variable with mean $\mu$ and variance $\sigma^2$. What is the upper-bound on the variance of $Y=\left|X\right|$? My gut feeling says that $\operatorname{Var}(Y) \leq \operatorname{...
-1
votes
1answer
2k views

conjugate prior for exponential distribution [duplicate]

If there is an exponential distribution $$p(x | \theta) = \theta\,e^{-x\theta}\mathbb{I}_{x>0}\, ,$$ what is a good conjugate prior? Also, will the posterior mean is a convex combination of prior ...
1
vote
0answers
140 views

Realistic Test Data Generation

I am searching for some literature or references, or also only some tips for the topic of test data generation that is realistic and what it means to the data to be realistic. Example Questions that ...
1
vote
1answer
46 views

PDF of 'blind' multinomial distribution

I want to describe the distribution where different coloured balls are drawn from a bag with replacement (so far, I know this is the multinomial). However, the observer only knows that he got ...
2
votes
1answer
412 views

Correlation between non-central chi squared & Gaussian

Is there an analytical result for the correlation between a (non-central) chi-squared distribution (parameters $d, \lambda$) and a standard Gaussian? More practically, I'm looking to sample two ...
0
votes
1answer
307 views

How to measure the difference of a distribution being normally distributed

Imagine I have a distribution like the following File SkewedDistribution.png of Wikimedia Commons by User:Audriusa licensed under CC-BY-SA 3.0 Now I want to measure, how this distribution differs ...
9
votes
3answers
12k views

A normal divided by the $\sqrt{\chi^2(s)/s}$ gives you a t-distribution — proof

let $Z \sim N(0,1)$ and $W \sim \chi^2(s)$. If $Z$ and $W$ are independently distributed then the variable $Y = \frac{Z}{\sqrt{W/s}}$ follows a $t$ distribution with degrees of freedom $s$. I am ...
0
votes
2answers
128 views

Lifetime or Failure Time

Lifetime / Survival time / Failure time : the time to the occurrence of event (always nonnegative) . Lifetime and Survival time can be synonymous . ...
2
votes
2answers
209 views

asymptotic distribution of a statistic

Say we have iid sample of size $n$ with $X_i \sim Exp(\lambda)$ and the task is to find asymptotic distribution of the statistic $$T_n := \frac{\bar{X}}{s}$$, where $s^2$ is the unbiased sample ...
1
vote
3answers
58 views

Inference about parameter $\theta$ be same?

Let $\mathbf x$ be a sample point and $T(\mathbf x)$ be a statistic of $\mathbf x$. Similarly, let $\mathbf y$ be a sample point and $T(\mathbf y)$ be a statistic of $\mathbf y$. In the book ...
6
votes
3answers
291 views

Transformation of Random Variable - Normal Distribution

Let $X$ be one observation from a $N(0,\sigma^2)$ population . What is the distribution of norm of $X$, i.e., $|X|$ ? My attempt : $$f_X(x;0,\sigma^2)=\frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{x^2}{2\...
1
vote
2answers
140 views

Ratio of CDFs $F(x)/x$ property

In my research project is useful to classify cumulative distributions functions of random variables with support in $[a,b]$ with $0\le a<b\le+\infty$ depending on whether the ratio, $$\dfrac{F(x)}{...
1
vote
1answer
94 views

References to papers/books that uses a kernel to smooth a discrete distribution

Since a kernel, such as Gaussian, is often used to smooth out the distribution of discrete points in 1D, 2D or 3D, I believe there must be some study materials or research work that have used this, ...
4
votes
2answers
271 views

Comprehensive list of distributions?

Is there a comprehensive list of distributions, e.g. gamma, Poisson, Gaussian, and when you should use each somewhere? My internet searching has been fruitless.
3
votes
1answer
222 views

Any practical uses of inverse uniform distribution?

To motivate a paper in game-theory I need examples of real-life uses of the inverse uniform distribution (http://en.wikipedia.org/wiki/Inverse_distribution#Inverse_uniform_distribution). Which type of ...
22
votes
5answers
6k views

What to learn after Casella & Berger?

I am a pure math grad student with little background in applied mathematics. Since last fall I have been taking classes on Casella & Berger's book, and I have finished hundreds (230+) of pages of ...
5
votes
1answer
516 views

Why is a symmetric distribution sufficient for the sample mean and variance to be uncorrelated?

While reading, I came across the puzzling statement that the sample mean and variance are uncorrelated only in symmetric distributions and there is strong correlation if the distribution is heavily ...
2
votes
2answers
154 views

Book/Website recommentation on probability distributions usage

Is there a book(s) / Website(s) that list the different type of distributions and real world usage or application for each one of them (beyond the typical know ones such as normal, binomial, poisson, ...
1
vote
0answers
55 views

How do I address a known bias in my sample?

I have a population of interest ($N = 5000$) for which I know some demographic information. I have a sample 1500 members of that population. So I have a good sized sample, and I know exactly how it ...