# 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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### 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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### 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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### 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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### 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 ...
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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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### 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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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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,...