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        Questions tagged [pr.probability]

        Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

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        23 views

        Bounding integral arising from expectation of a random variable satisfying Bernstein's inequality

        Let $X$ be a random variable s.t. for $v, b > 0$ and $C \geq 1$: $$ P(X \geq t) \leq C\exp\left(-\frac{t^2}{2(v^2 + bt)} \right) $$ I am trying to show that $\mathbb{E}X \leq 2v(\sqrt{\pi} + \...
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        Numerical expectation involving Dirac-delta function

        I'm looking for the way for numerical integration including Dirac-delta function. Here is what I want to obtain in numerical way such as Monte Carlo sampling. $$ \int m(\mathbf{x})\delta(G(\mathbf{x}))...
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        32 views

        Computational Time Complexity bounds for approximate maximum of a sequence/array

        The problem I have is the following: Given a sequence $x_1, \ldots, x_N$ for $N$ very large. For any $\varepsilon, \delta > 0$, find a number $\hat{x}_{\varepsilon, \delta}$ such that $$\mathbb{P}(...
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        74 views

        How to find a special random variable?

        Suppose random variables $X_1$ and $X_2$ have the same distribution under P, $Y_1$ is an arbitrary random variable,let $Z_1:=X_1+Y_1$.Can we find a r.v. $Y_2$ which has same distribution as $Y_1$,such ...
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        Joint PDF of Laplace distribution and Gaussian Distribution

        If $X$ follows a Laplace distribution with PDF $f(x\left| {\mu ,b} \right.) = \frac{1}{{2b}}\exp \left( { - \frac{{\left| {x - \mu } \right|}}{b}} \right)$ where $\mu$ is a location parameter and $b&...
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        Skorokhod representation for weak convergence of exchangeable arrays

        Let $(X_e)=(X_e)_{e\in \mathbb{N}^{(k)}}$ be a $k$-dimensional exchangeable real random array (see this note for the definition), where $k\in \{1,2,\ldots \}$ is fixed and $\mathbb{N}^{(k)}$ denotes ...
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        Expected size of matchings in a cubic graph

        Let $G$ be a random cubic graph on $n$ vertices. Let $M$ be the set of (not necessarily maximum) matchings of $G$. What is the expected size (i.e. number of edges) of an element of $M$? In other ...
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        Mixing time and spectral gap for a special stochastic matrix

        Conisder the following dimension stochastic matrix, \begin{bmatrix} p & q & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 &...
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        Theorems like the Lovász Local Lemma?

        The Lovász Local Lemma gives a probability bound in a context where there are many events that are "not quite" independent. What other theorems exist in this genre? That is, what other theorems have ...
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        1answer
        75 views

        concentration inequality for a weighted sum of independent but not identical binary variables

        Let $\alpha\in[0,1]$ be a fixed constant, and let $w,x\in[0,1]^n$ be two vectors such that $\sum_i w_i x_i=\alpha$. Define $Y = \sum_i w_i X_i$, where $X_i \sim \operatorname{Bernoulli}(x_i)$, so it ...
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        Monotonicity of $\mathbb{P}\left[\sum_{i=1}^n X_i \leq n \cdot c \right]$ in $n$ for $c>\mathbb{E}(X).$

        The nice question below was answered in the affirmative in On the sum of uniform independent random variables Let $X_1,...,X_n$ be independent uniform random variables in [0,1] and assume $c>1/2$. ...
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        How to calculate the expectation of N? [on hold]

        How to calculate the expectation of N? X is a random variable with an uniform distribution in the range of [0,1], Z=X*X, Y1,Y2,Y3...Yn are independent and identical distributed random variables,Yi ...
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        1answer
        58 views

        Asymptotic bound on the total variation distance between a standard multivariate normal and a simple mixture

        Let $P = N(\vec{0}, I^d)$ be a standard multivariate Gaussian distribution in $d$ dimensions. Let $Q$ be distributed the same as $P$, except that samples from $Q$ have one of their coordinates, chosen ...
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        About a class of expectations

        Consider being given a $n-$dimensional random vector with a distribution ${\cal D}$, vectors $a \in \mathbb{R}^k$, $\{ b_i \in \mathbb{R}^n \}_{i=1}^k$ and non-linear Lipschitz functions, $f_1,f_2 : \...
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        2answers
        137 views

        Integral formula involving Legendre polynomial

        I would like a proof of the following equation below, where $(P_n)$ denotes Legendre polynomials. I guess this formula to hold; I checked the first several values. \begin{equation} \int_{-1}^{1}\sqrt{...

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