# 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.

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### General distributions with the “transportation-cost inequality” property to piece log-concave distributions

It is now known [Otto et Villani 2000; Cordero et al 2006; etc.] that on an $n$-dimensional smooth Riemannian manifold $X$ and a probability measure $\mu$ on $X$ with density $d\mu \propto e^{-V}dvol$ ...
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### Maximum Number of modes of $V=U+Z$ where $Z$ standard normal and $|U|\le a$

Let $f_V$ be a pdf of random variable $V$ where \begin{align} V=U+Z \end{align} and where $U$ and $Z$ are independent and $Z$ is Gaussian. Moreover, suppose that $|U| \le A$. Can we find the upper ...
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### Closed form of :$\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx$ for $k$ is even integer and :$\int _{0}^{t}\exp(-x^2 \operatorname{erf}(x))dx$

This question is related to my question here such that i want to find a closed form of $\int_{-1}^1 x^{2k} (\operatorname{erf}(x))^k \,dx$ , for $k$ is even integer because for odd integer is $0$ as ...
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### Reference request: discretisation of probability measures on $\mathbb R^d$

Given a probability measures $\mu$ on $\mathbb R^d$ with finite first movement, i.e. $$\int_{\mathbb R^d}|x|\mu(dx)~~<~~+\infty.$$ My concern is to approximate $\mu$ some $\mu_n$ that is ...