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        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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        93 views

        The norm of isotropic sub-Gaussian random vector may not be sub-Gaussian

        Suppose $X$ is a isotropic sub-Gaussian $n$-dimensional random vector (i.e. $EXX^T=I_n$, and for any unit vector $u$,$\|\left<X,u\right>\|_{\psi_2}\le K$). It is said that $\|X\|_2-\sqrt n$ may ...
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        2answers
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        Is there a name for “splitting a probability distribution into independent components”?

        Suppose I have a random variable $\theta=(\theta_1,\dotsc,\theta_n)$; where the $\theta_i$ might have pairwise correlations. I decompose it into $\theta=\hat\theta(\phi_1,\dotsc,\phi_k)$, where $\hat\...
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        29 views

        Distribution of a post-selected random variable with power-law distribution

        Background Assume $X \sim \mathcal D$ is a random variable, distributed according to some distribution $\mathcal D$. Then postselection with respect to some set $A$ is defined as the conditional ...
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        Reference Request: Total Variation Between Dependent and Independent Bernoulli Processes

        Let $X$ be a random variable taking values in $\{0,1\}^n$ with the following distribution. For each coordinate $i$, we have $p_i = P(X_i = 1) = c/\sqrt n$, where $c$ is a (very small) constant. ...
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        50 views

        Order statistics of correlated bivariate Gaussian

        Suppose $(X_1,Y_1),...,(X_n,Y_n)$ are i.i.d. bivariate Gaussian with mean zero. Each coordinate has variance 1 and correlation between coordinates is $\rho\in[-1,1]$. I'm interested in the following ...
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        Strictly Proper Scoring Rules and f-Divergences

        Let $S$ be a scoring rule for probability functions. Define $EXP_{S}(Q|P) = \sum \limits_{w} P(w)S(Q, w)$. Say that $S$ is striclty proper if and only if $P$ always minimises $EXP_{S}(Q|P)$ as a ...
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        A clean upper bound for the expectation of a function of a binomial random variable

        I wonder if there is a closed-form, or clean upper bound of this quantity: $\mathbb{E}[|X/n-p|]$, where $X\sim B(n,p)$.
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        How to find a special random variable? [closed]

        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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        1answer
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        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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        Looking for a generalization of Binomial distribution and it's properties

        In my research (coming from computer science), I have encountered a family of discrete probability distributions that seems to be some sort of generalization of the binomial distribution. A ...
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        1answer
        52 views

        Concentration inequality for quadratic form of Gaussian variables with non-idempotent matrix

        Given $y \sim N(0,\sigma^2 I)$, and $M$ that is a symmetric matrix (not necessarily idempotent) what is the distribution of ${y^T M y}$? is there a high probability bound on $|{y^T M y}|$? Most ...
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        1answer
        51 views

        Rate of decay in the multivariate Central Limit Theorem

        The celebrated Berry-Esseen inequality tells us that the rate of convergence in the univariate CLT is of magnitude $\frac{1}{\sqrt{n}}$ for sums $S_n=X_1+\cdots+X_n$ of independent random variables $...
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        Variance of sum of $m$ dependent random variables

        I originally posted this question in Mathstackexchange, but since I got no answer I'm posting it also here. Let $X_1,X_2,...$ be a sequence of identically distributed and $m$-dependent random ...
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        Different balls in bins: What is the probability distribution of the sum of the minimum of the two types of balls over all bins?

        Assume that there are $N$ different bins and two different kinds of balls, $R$ red balls and $W$ white balls. The red balls and the white balls are randomly distributed across the bins (that is, for ...

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