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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Convexity of truncated expectation

Let $k, n$ be two positive integers with $k \leq n$, and let $P = \{ (x_1, \dots, x_n) \in [0, 1]^n : \sum_i x_i = k \}$. Given $x = (x_1, x_2, \dots, x_n) \in P$, let $X_i$ be the random variable ...
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28 views

Probability problem [on hold]

In an urn there are 8 blue and 5 red balls. Consequently, we are taking 1 ball without returning it, until all of one of the colors are taken out. Let X is the amount of balls. Find the type of ...
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21 views

Expectation of two identical log-normal distributions [migrated]

I would like to compute the conditional expectation (on an interval from $c$ to $\infty$) of the minimum of two log normal distributions. Denote $X_1$, $X_2 \sim LN(0, \sigma)$, the associated ...
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15 views

Product of lognormal random variables

Let $X_1$ and $X_2$ be two normal random variables. Write $X_1\sim N(\mu_1, \sigma^2_1)$ and $X_2\sim N(\mu_2, \sigma^2_2)$, to fix ideas. Consider the corresponding log-normal random variables: ...
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38 views

Characterize Linear Transformation of Dirichlet Distribution

Let $X=(X_1,....,X_K)\sim{}\text{Dir}(\alpha_1,...,\alpha_K)$ be a Dirichlet distribution with parameters $\alpha_1,...,\alpha_K$. Let $A$ be a non-singular linear map and ...
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35 views

Proving Fixed Point Algorithms

In Thomas Minka's paper on Estimating the Dirichlet Distribution (link here http://research.microsoft.com/en-us/um/people/minka/papers/dirichlet/minka-dirichlet.pdf), the author presents a fixed ...
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15 views

Distribution of deviations from order statistics [migrated]

Consider a continuous r.v. $x$ with CDF $F(\cdot)$. Let $\{x_i,\,i=1,\ldots,n\}$ be a sample of $n$ IID draws, and let $\{x_{(i)},\,i=1,\ldots,n\}$ denote the order statistics; i.e., $x_{(1)}$ is the ...
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35 views

Existence of probability distribution satisfying upper/lower bounds on events

Suppose we have a finite sample space $S$ and some events $A_1, \dots, A_k \subseteq S$. We would like to put a probability distribution on $S$ so that no element has probability greater than a ...
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60 views

The role of absolute continuity in stochastic ordering defined over sets of probability distributions

This question is about a claim given in this paper (page 261, the remark), but without any proof. It simply says that if two sets of probability distributions, $\mathscr{P}_0$ and $\mathscr{P}_1$ ...
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28 views

On the numerical range of non-self adjoint Gaussian matrix

For a complex $n \times n$ matrix $A$, its numerical range is the set $$W(A) = \left\{\mathbf{x}^*A\mathbf{x} \mid \mathbf{x}\in\mathbb{C}^n,\ \|x\|_2=1\right\} .$$ We can further define the ...
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68 views

On the eigenvalues' distribution of random unitary

Fix an integer $d$, let $\mathbb{U}_d$ be the $d\times d$ unitary group. For any $U\in \mathbb{U}_d$, define $\Omega(U)$ be the length of the smallest arc containing all the eigenvalues of $U$ on the ...
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158 views

Expected value and variance of a stochastic process

I would like to ask if there is a way to find the expected value and the variance of the following process $$ dv_t=(a-be^{\alpha v_t})dt+\sigma dW_t, \quad v_t=v_0 $$ where $a\in (-\infty,+\infty), ...
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1answer
131 views

Upper Bound for the Difference of Even Probability and Odd Probability in Hypergeometric Distribution

Let $X$ be a random variable following the hypergeometric distribution with parameters $N,K,n$, where \begin{equation} Pr(X=k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}. \end{equation} To ...
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59 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 ...
2
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1answer
66 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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27 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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1answer
212 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 ...
3
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0answers
31 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 ...
5
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2answers
141 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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49 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
49 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
101 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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40 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 ...
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87 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
122 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
119 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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1answer
149 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
119 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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50 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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61 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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30 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
179 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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28 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: ...
3
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1answer
362 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 ...
3
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1answer
169 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: $$ ...
2
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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
132 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
105 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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52 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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34 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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46 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 $X$ ...
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78 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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277 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
114 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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66 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 ...
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1answer
95 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
33 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 ...