# Questions tagged [probability-distributions]

In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.

981 questions
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### Distribution of largest entry in a random vector

If we have a random unit vector on $\mathbb{C}^n$, drawn from the Fubini-Study metric, the marginal distribution of the squared absolute values of each of the coefficients in the vector is given by a ...
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### Optimization of generalized Fisher information over space of probability measures

Assume we already have $p(\theta)$ apriori desnsity function about unknown parameter $\theta$, and we consider an optimization problem as follows: \max_{P \in \mathcal{M}} \int_{\...
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### Compounding two Dirichlet distributions

If $\pmb y|\pmb x\sim\text{Dir}(\alpha\pmb x)$ and $\pmb z|\pmb y\sim\text{Dir}(\beta\pmb y)$, then what can be said about the distribution of $\pmb z|\pmb x$? For example, is it possible to derive ...
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### Bounding quantiles of the noncentral chi distribution

I need to bound the empirical quantiles for a noncentral chi distribution (not chi-squared) $\chi_\nu(\lambda)$, where $\nu$ is the number of degrees of freedom and $\lambda$ the non-centrality ...
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### In Wasserstein, what is the relationship between $W_2 (\widehat{\mathbb{P}}_{N},\mathbb{P})$ and $W_2 (\widehat{\mathbb{P}}_{N}^{x},\mathbb{P}^{x})$?

Before presenting my question (which I already formulate in the title of this post) is important to establish the context of my problem: Definition: The $p$-Wasserstein metric $W_{p}(\mu,\nu)$ ...
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### Calculate Average and Correlation of WSS Random Processes

Given two stochastic processes, $X[n]$ and $Y[n]$, both being WSS (wide state stationary) and independents. What would be the Average and Autocorrelation function of $Z[n] = Y[n] X[n]$? Is the ...
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### $p$-th moment of complex Gaussian random variable

Let $1<p<2.$ Let $G$ be a complex Gaussian random variable. then what is the value of $\mathbb{E}[|G|^p]$ ? The symbol $\mathbb{E}$ denotes the expectation of a random variable.
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### Berry-Esseen type theorem for Monotonic independence

The central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample converges to a normal distribution as the sample size increases ...
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### Calculate the expectation of the maximum of averaged random walks

Let $X_1, X_2, \ldots$ be iid random variables with bounded second moment. The question is to calculate the exact value of $$\mathbb{E} \max_{1 \le j < \infty} \frac{X_1 + \cdots + X_j}{j}.$$ Is ...
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### References for the theory of Hadamard functions and compositions of random vectors

Recently, I fell in love with the pointwise/elementwise/componentwise/Hadamard/Schur functions and compositions of random vectors such as Hadamard squares and products of random vectors. Here is one ...
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### Probability distributions with all positive cumulants

Is there a term for a distribution with all cumulants positive (or nonnegative)?
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### Asymptotics of the joint pdf of two sums of powers of independent $\mathcal U(0,1)$ random variables

As a warm-up in words: The sum of twelve uniform random variables is a classic approximation to a normal distribution. What is the joint pdf for the sum of their cubes and the sum of their fourth ...
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### maximal distance of nearby iid unifrom random variables

Question: Let $X_1, \ldots ,X_n$ be $n$ iid uniformly distributed random variables, i.e., $X_j \sim \mathcal{U}(0,1)$ for each $j=1,\ldots ,n$. What is the PDF of the maximal distance between to ...
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