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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Probability that one normal (uncorrelated) variable is greater than another if the latter is positive [on hold]

Assume that $X\sim N(0,\sigma_x^2)$, $Y\sim N(0,\sigma_y^2)$ and $X$ and $Y$ are uncorrelated. Can we solve analytically for $\mathbb P(X>Y |Y>0)$?
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34 views

Multimodal property of polynomial logistic distribution

Let $P(x)$ be a polynomial (of an odd degree $n$) strictly increasing on $(-\infty, +\infty).$ Then $F(x)=\displaystyle \frac{1}{1+\exp\{-P(x)\}}$ is a distribution function of a polynomial logistic ...
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1answer
57 views

Median of a uniform multinomial variable

Let $k\in\mathbb N^+$ be a positive integer. Consider a set of i.i.d. random variables $X_1,X_2,\ldots, X_n$, each of which is distributed uniformly over $\{1,2,\ldots,2k+1\}$. For $i\in ...
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21 views

Expected number of perfect matchings in bounded degree bipartite graphs

Consider collection $\mathcal C_{n,n,\Delta}$ of every $2n$ vertex balanced bipartite graph of average degree $\Delta$. What is the expected number of perfect matching a graph in $\mathcal ...
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8 views

Sampling distribution with upper and lower limit [closed]

I want to sample from my data and plot a distribution of means, however they are percentages which only fall within 0-100, therefore i can not use central limit theorem because they are not normal ...
5
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1answer
206 views

Estimate of incomplete binomial integral

Let $0\le k \le n$. Prove that $$ n\binom{n}{k}\int_{0}^{\frac{k}{n+1}}t^k(1-t)^{n-k}\,dt \le 1/2. $$ As far as I know 1) it is proved for $\frac{k}{n+1}\le 1/2$ and 2) not proved for $1/2 ...
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28 views

Joint distribution on order statistics and sample history

If samples $X_1, X_2, ... X_t$ are picked independently and identically from the discrete uniform distribution $[1,2, ..., P]$, what is the joint distribution of the last $k$ order statistics and last ...
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19 views

Check uniformly distributed continuous random process [migrated]

I have a random process which I know is uniformly distributed, and I expect it to be distributed between the range $[0,1]$. Then, I generate 100 realizations of the process. The question: Which is ...
5
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2answers
133 views

Expected number of changes in the sign of a rolling sum of independent normal variables

Imagine we define $Y(t+n)= X(t+1)+.....+X(t+n)$ where $X(i)$ is an independent normal (i.e. everyday we remove the starting observation and we add a new one). We have $n$ consecutive observations of ...
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44 views

BM hitting times with exponential killing process

Assume a BM in 3d domain (infinite) with a small absorbing subdomain (cube, sphere, ect), centered at point $p_s=(x_s,y_s,z_s)$ . BM starts at point $p_0=(x_0,y_0,z_0)$ and when it riches the ...
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1answer
44 views

Fundamental difference between Poisson Point Process and Binomial Point Process

What is the fundamental difference between Poisson Point Process and Binomial Point Process? I am evaluating a solution in a Binomial Point Process setup. If I want to evaluate that in a Poisson ...
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1answer
88 views

Choosing a sample based on where the density function is highest

Is there a name for the following process? Say I have an absolutely continuous probability density function $f$ with compact support, and I take $k$ independent samples $x_1,\dots,x_k$ from $f$. ...
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38 views

A question about probabilistic graphical models

Say one is given a probabilistic graphical model and a cut of the underlying graph. Do we know any statements about when and how can one or many of the marginals (of the sources) or the conditionals ...
2
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0answers
86 views

Growth of inner products between two random vectors on the sparse hypercube

We define the $s$-sparse hypercube in $\mathbb{R}^d$ as \begin{align} \mathbb{H}_s = \bigl \{ {\bf{v}} \in \{ -1, 0 , 1\}^d \colon \| {\bf{v}} \|_0 = s \bigr\}, \end{align} where $ \| {\bf v} \|_0 $ ...
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1answer
115 views

Counterexample: weak convergence doesn't imply $L^1-$convergence [closed]

I'm not sure my question is of research level, but I cannot find the answer in the existing reference. Let $\mu_n$ be a sequence of probability measures on $\mathbb R$ satisfying $$\int_{\mathbb ...
2
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1answer
113 views

About Renyi entropy

If one is given a joint probability distribution over a finite set of discrete random variables then I guess there a notion of $\alpha-$Renyi entropy defined for it as $S_\alpha (X_1,..,X_n) = ...
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69 views

Finding a closed form for a certain double integral

I am working with bivariate and accurate Birnbaum-Saunders distribution to find the probability density function of a particular model for this, I would like to find a closed form for the full below: ...
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1answer
148 views

Averaged geometric series with floor function

Given a value $p\in[0,1]$ (a probability of occurrence), I would like to bound the following expression: $$ s\frac{1-(1-p)^{k+1}}{p(k+1)} + (1-s)\frac{1-(1-p)^{k}}{pk},\ \ \ \text{where $k=\lfloor ...
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175 views

A contractive mapping which I don't understand

Given a matrix $Y$ and a vector $c$ define the following iteration $\hat{c} = f(c)$, where each element of $\hat{c}$ is given by $$\hat{c}_{\ell} = \frac{\sum_k ...
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1answer
116 views

Neat definition of Harris Ergodicity

I can't find any reference where the definition of Harris Ergodicity for Continuous time Markov processes is defined. a) What would be exactly the definition? b) What reference could be helpful? ...
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1answer
115 views

Is Gaussian the unique 2-stable distribution? [closed]

It is well known that Gaussian distribution is a 2-stable distribution. (For more information about p-stable distribution, please refer to Stable Distribution.) But is Gaussian the unique 2-stable ...
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49 views

variance of log of ratio of chi-square variables

Let X be a chi-square variable with two degrees of freedom. Let A and B be to arbitrary constants, with $A>B>0$. I need the variance of $Y=\log(1+AX)-\log(1+BX).$ The mean is, maybe not ...
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58 views

Algorithm to calculate moments of uniform distribution on convex polyhedra

There is system of linear inequalities $$ Ax \leq K, $$ $$ x\geq a, x\leq b. $$ $A$ is $(n\times m)$-matrix, where $n\approx 100$ and $m\approx 10000$, $rank(A)=n$. Suppose that on set of solutions ...
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24 views

Distribution of stopped Brownian motion in $\mathbb R^2$

Let $B=(B^1_t,B^2_t)_{t\ge 0}$ be a standard Brownian motion in $\mathbb R^2$. Let $U=(U^1,U^2)$ be an independent random variable taking values in a circle $C_1\subset\mathbb R^2$ with uniform ...
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1answer
148 views

connection between the statistical properties of a scalar field and its columns

Consider a scalar field $s:[0,1]^3 \to \mathbb{R}$ and its "column" field \begin{equation} c: [0,1]^2 \to \mathbb{R}: (x,y) \mapsto \int_0^1 s(x,y,z) \,\mathrm{d}z. \end{equation}. What can be said ...
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27 views

derivation of a gap related to extreme value theory

I have an expression to evaluate as follow: $\mathbb{E}\left[\sum_{k=1}^K s_k f(x_k)\Big|s_k=s_k^{\ast} \right]$ where $\{s_k^\ast\}$ can be treated as a ${policy}$ which is defined as follows: ...
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1answer
281 views

Bounds on $\int \log(1+x) g(x) \mathrm{d}x$?

Let $X$ and $Y$ be two continuous real random variables with common support $(0,x_{\max}]$ and with PDF $f_X(x)$ and $f_Y(y)$. Assume that $\Pr [Y\geq\beta \mid X<\beta] \leq k$ and that $\Pr ...
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20 views

How would I derive the Hellinger distance from the Hellinger integral?

What function would I integrate to derive the following expression: $d(H_1,H_2) = \sqrt{1 - \frac{1}{\sqrt{\bar{H_1} \bar{H_2} N^2}} \sum_I \sqrt{H_1(I) \cdot H_2(I)}}\\$ I understand that this is ...
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1answer
167 views

Solving recursion / finding generating function of a probability mass function

I am assessing the probability distribution on a running time of some algorithm that we've developed. I am looking for a family of probability mass functions $f_n$ with the following recurrence: $$ ...
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2answers
121 views

Difference between maxima of random variables

Given four independent, identically distributed Gaussian random variables with zero mean and unit variance $x_1$, $x_2$, $y_1$, $y_2$, consider \begin{equation} u \equiv \max(x_1+C\, y_1, x_2+C \, ...
3
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1answer
126 views

Moment matching on the standard simplex

Let $\vec{\mu}_1, \vec{\mu}_2,\ldots, \vec{\mu}_k \in \Delta^{d-1}$ be $k\ (k\geq 2)$ distinct vectors on the standard simplex, where $$\Delta^{d-1} = \{\vec{\mu}\in R^{d}:\| \vec{\mu}\|_1 = 1,\mu_j ...
3
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1answer
102 views

Learn a distribution from distributions on samples

There's many good ways to learn a distribution $p_X$ of an r.v. $X$ over $k$ symbols given many i.i.d. samples $X_1,\ldots, X_n$. The simplest is to use the sample relative frequencies $\hat{f}_X$ as ...
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50 views

Spectrum of sum of fixed matrices with random signs

Let $A_1,\ldots,A_k$ be a given sequence of $N$-by-$N$ Hermitian matrices. Assume all have spectrum contained in $[-1,-\delta] \cup [+\delta,+1]$ for some $\delta>0$. Let $$A=\frac{1}{\sqrt{k}} ...
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33 views

A functional's expectation using both known and unknown pdf

Suppose we have a random variable $X$ with a known distribution $f$ over an interval $[a,b]$ and another r.v $Y$ over the same interval but with an unknown distribution $g$. We also have a functional ...
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45 views

Stochastic Ordering of Negative Binomial-like Distributions

Please forgive me if this is not precise enough to post here. Simply ask me to remove it if it is not suitable. I am new here. I am bounding the running time of an algorithm as a random variable $X$ ...
3
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68 views

How does Jensen Shannon divergence and KL divergence correlate?

I am wondering if there is way to derive the correlation between Jensen Shannon divergence and KL divergence for two distributions: P and Q, in order to show that if JSD(P,Q) decreases, KLD(P,Q) ...
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1answer
273 views

Berry-Esseen bound for martingale sequence with varying and dependent variances

Let $(X_{1},\ldots,X_{k},\ldots)$ be a martingale difference sequence, i.e. $$ E[X_{k}|\mathcal{F}_{k-1}] = 0 $$ where $\mathcal{F}_{k-1}$ is the $\sigma$-algebra filtration at $k-1$. Let ...
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1answer
99 views

A graph assignment problem

Consider bipartite graph with vertex set $V_1\cup V_2$ where $|V_1|=\frac{n(n-1)}2$ and $|V_2|=n$. The vertices in $V_1$ all have degree $2$ and connected to two vertices in $V_2$. The vertices in ...
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63 views

Proof of variance in wishart distribution

I wanna to prove the variance of wishart distribution, first a brief description of wishart distribution, how can i proof it? I wrote a solution but the result is not correct, please help me to fix ...
3
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1answer
93 views

expected value of multiplication of matrices

I start with background and then ask my question, background is a brief description of wishart distribution. Background The Wishart distribution with $\nu$ degrees of freedom and positive definite ...
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1answer
32 views

how to resolve the infinite nesting of interactive POMDP

I am reading papers about I-POMDP. I cant understand the finitely nested I-POMDPs given in these papers. The belief update of the algorithm has a problem that agents' belief updates mutually depend ...
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181 views

universality for large deviations?

This is a question about universality in probability theory, with combinatorics in mind. Consider a sequence of polynomials $P_n$ in one variable, with positive coefficients. Combinatorics is a large ...
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25 views

Hypergeometric distribution with a priori probabilities of the balls

If we have an urn with $N$ balls of two colours ($D$ red and $N-D$ black balls respectively), then the probability of having $k$ red out of $n$ balls drawn at once without replacement follows the ...
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32 views

Processes with the same finite dimensional distributions as the solutions to SDEs

Consider a sequence of stochastic processes $\{\tilde{x}^n\}$, $\tilde{x}^n = \tilde{x}^n_t(\omega)$, and Brownian motions $\{\tilde{w}^n\}$. Suppose that for each $\tilde{x}^n$ solves the stochastic ...
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1answer
178 views

Supremum of a martingale

Let $(X_n)$ be a martingale. What can be said about the distribution of its maximum over a window of fixed length: $$M_n = \max_{n-10 \leq k \leq n} X_k$$ or about the "range" over a window: $$R_n = ...
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2answers
174 views

Gaussian and the convex hull of moment curves

Let $c_1,\dots, c_d$ be the first $d$ moments of the standard normal distribution. Does the point $(c_1,\dots, c_d)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a ...
5
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2answers
125 views

Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian

This is a question related to the statistical model behind independent component analysis (ICA). We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a ...
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1answer
89 views

Distribution of maximum unique number of several random numbers

Suppose discrete random variables $\{X_1, X_2, ..., X_n\}$ are i.i.d. described by the probability function: $f(x) \equiv \text{Pr}(X_i = x)$, and $X_i \in \{1,2,3, ..., m\}$. Let $Y$ be the ...
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1answer
79 views

Question about the weak convergence of probability

Let $\mu$ be a probability measure on $\mathbb R$ and set $$c(K):=\int_{\mathbb R}(x-K)^+d\mu(x).$$ Assume that one has a sequence of probability measures $(\mu_n)_{n\ge 1}$ s.t. $$\int_{\mathbb ...
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1answer
115 views

Question abouth Skorokhod representation of random variables (II)

This is a continuation of Question abouth Skorokhod representation of random variables Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ such that $$\int_{\mathbb R}|x|^pd\mu(x),~ ...