Questions tagged [distributions]

A distribution is a mathematical description of probabilities or frequencies.

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16
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607 views

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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267 views

Are there non-trivial settings where the MAD statistic has a closed-form density?

The MAD statistic of an iid sample $(x_1,\ldots,x_n)$ is defined as the median of the absolute deviation from the median: $$ \text{mad}(x_1,\ldots,x_n)=\text{med}\left\{|x_i-\text{med}(x_1,\ldots,x_n)|...
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3k views

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, say I have a group of customers and have calculated their spend over a fixed length of ...
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321 views

Distribution with a given moment generating function

As a follow-up to a question on a central limit theorem for independent random variables (r.v.) here, let $Y_j=-\log(1-V_j)$, where $V_j\sim\mbox{beta}(1-\sigma,j\sigma)$, $j\in\mathbb{N}^*$, $\sigma\...
7
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504 views

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|$. $Y=|X|$ ...
7
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1answer
189 views

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 ...
6
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101 views

Moments of $Y=X_1 + X_2 X_3 + X_4 X_5 X_6 +\cdots$

The $X_i$'s are i.i.d. and $X$ denotes any of these random variables. We assume here that $|E(X)|<1$ to guarantee convergence. I am interested in particular in the third moment $E(Y^3)$. For the ...
6
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914 views

What does it mean for a probability distribution to not have a density function?

I understand the distinction between probability mass and density functions. But I don't understand what it means for a continuous random variable to have a probability distribution but not a density. ...
6
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45 views

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, ...
6
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709 views

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. ...
6
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226 views

Multinomial distribution conditional on number of distinct items

I want to sample from the integers $\{1, \dots, k\}$ with probabilities $\{ p_i \}_{i=1}^k$, with replacement, until I see $m$ distinct elements (call that $n$ times). You can view the distribution I ...
6
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260 views

Sampling distribution of sample trimmed (truncated) mean

It is elementary probability theory that the sample mean of an i.i.d. sample follows normal distribution, if the background distribution is normal. But what about the trimmed mean? Is there any result ...
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129 views

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 ...
6
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138 views

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 $\frac{b}{b+w}$ ...
5
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107 views

CDF that combines properties of Pareto and Exponential

Let $Y$ be a random variable defined on the domain $[1;\infty)$ that is distributed according to the cdf $G_Y(y)$. A Pareto distribution, $$ G_Y(y) = 1 - y^{-\theta}$$ has the property that $$ P(Y&...
5
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81 views

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 ...
5
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225 views

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 $Z = X_1 + X_1 X_2 + X_1 X_2 X_3 +\cdots$ ...
5
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200 views

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 $\{x_1, \dots, x_n\}$...
5
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74 views

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 ...
5
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36 views

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 ...
5
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130 views

Finite sum of beta prime iid random variables

The beta prime distribution is infinitely divisible, as proved in Steutel and van Harn, 2003 (Appendix B). Sadly, in this book, there is no espression of the parameters of the distribution of n ...
5
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285 views

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 $F,$ $MF(t),$ ...
5
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1answer
78 views

Functions of continuous random variables

Let Y be an exponential random variable with parameter $\tau > 0$. Compute the cdf and pdf of $F_W$ where $W = Y^3$ The solution states the cdf as $1 - e^{\frac{-y^\frac{1}{3}}{t}}$ because $F_Y =...
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507 views

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 $\...
5
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145 views

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 ...
5
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232 views

Probability distribution models compatible with quantile regression

I am familiarizing myself with quantile regression. I understand it is first and foremost an estimation method such as e.g. OLS. But I wonder about the probability distribution models for which ...
5
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749 views

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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62 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 ...
5
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89 views

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 ...
5
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1k views

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 ...
5
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984 views

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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234 views

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 ...
5
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619 views

Equality vs. Equality in Distribution ($t$-distribution for example)

A technical question that came up to mind as I was reading up on linear models today. Consider the $t$-distribution with $\nu$ degrees of freedom ($t_\nu$) for example. Let's say $T \sim t_{\nu}$; ...
5
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183 views

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....
5
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2k views

Detecting outliers in non-normal distribution data

I'm working with data from a resistivity test. However, during the test it is common that a few measurement points are wrong due to technical failure. So I want to find and remove these points. I ...
5
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1answer
85 views

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. ...
5
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0answers
145 views

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 $F_n(x)=\frac{1}{n}\sum_{i=1}^n1_{\{X_i\leq x\}}$...
5
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350 views

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}+...+...
5
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0answers
84 views

Gamma distribution different derivations

According to this link - http://cnx.org/contents/2d28fe6a-5000-454e-a2b9-6fbca9e9b56c@3/THE_GAMMA_AND_CHI-SQUARE_DISTR the waiting time of the $k$th event in a poisson process is gamma distributed. ...
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909 views

How to find the distribution of the weighted sum of independent Bernoulli random variables for positive non-integer weights

How do I find the distribution of the weighted sum of independent Bernoulli random variables if the weights are non-negative real numbers? I have N number of independent Bernoulli distributed random ...
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0answers
463 views

Are there useful distributions for ternary variables (e.g. $-1,0,1$ data)?

The title says it. If one wishes to analyze ternary outcomes—that is categorical outcomes with specifically three values (-1,0,1)—are $\chi^{2}$ / contingency table tests and multinomial-logistic ...
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662 views

Sum of independent Wishart with same degrees of freedom but different scale matrices

Is there any result showing that a sum of independent Wishart with same degrees of freedom but different scale matrices is a Wishart? For example, if I have two random variables: $$ Y \sim W_p(n,\...
5
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0answers
2k views

How to get distribution of sum of dependent bernoulli variables

I have $N$ Bernoulli variables, $X_1,...,X_N$ and $X_i\sim B(1, \pi_i)$, $\pi$ is known for each $X_i$, and $Y=X_1+...+X_N$, now I need to get the destribution of $Y$. If $X_i$ and $X_j$ are ...
5
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0answers
993 views

The distribution of STD/MAD for a Student-t

Where $X \sim a$ symmetric Student-t Distribution $t_\alpha$, with power law tail $\alpha>2$, looking for the distribution of $$ \frac{\sqrt{ \sum_{i=1}^n x_i^2 }}{\sum_{i=1}^n |x_i|}, $$ in ...
5
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0answers
111 views

Zipf's vs Self-similar: are they really the same

Recently I ran into a test using both zipf's and self-similar generated datasets. I followed the description from Jim Gray's paper on generating such datasets (Quickly Generating Billon-Record ...
5
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1answer
339 views

Transforming two normal random variables

I'm reviewing for a test, and I am not sure if I am getting the right solution. Let $X$ and $Y$ be iid $\mathcal{N}(0, \sigma^2)$ random variables. a. Find the distribution of $U = X^2 + Y^2$, $V = \...
5
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0answers
132 views

Joint distribution of column sums when row sums are fixed

Suppose I have an $m$ by $n$ table $X_{ij} \in \{0,1\}$, where in each row, $r$ randomly chosen entries are set to 1 (the rest are 0), i.e. $\sum_j X_{ij}=r$. I know that e.g. the column sum $\sum_i ...
5
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67 views

How to define and model consumption bundles?

Imagine an a la carte buffet with n different rooms. On entering the buffet you pick a room (let's say American, Mexican or Italian food) where you stay for the duration of your visit. Once in a room,...
4
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82 views

How to determine the distribution of a parameter fit by nonlinear regression

The following (rectangular hyperbola) equation is used often in biology, often with additional terms (this is simplified). K is the Km of enzyme kinetics, the EC50 of pharmacology, etc. X is ...
4
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0answers
33 views

Which financial time series have a PDF and/or a CDF?

Consider the following types of financial time series for a single publicly-listed stock: Price data Log returns Cumulative returns Each is computed from the item listed before it: log returns are ...

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