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0
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0answers
38 views

Conditional probability of dependent random variables

Let $ X \sim f_X(x), Y \sim f_Y(y) $ are two dependent random variables and their corresponding PDFs. I want to find a probability $$ P(Y\ge 0 | X+Y\ge 0) .$$ If these variables were independent I'd ...
5
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0answers
112 views

Extrapolation between longest increasing and longest alternating subsequences

The question When should we expect Tracy-Widom? motivated me to post the following question, in which I have been interested for a while. Let $f(n)$ be a function from the positive integers to ...
0
votes
0answers
18 views

Inventory Distribution [closed]

I have inventory for goods that needed to be distributed to each branch as depicted in this picture http://i.stack.imgur.com/pxu8D.jpg and how much each branch will get goods depends on past sales ...
13
votes
2answers
361 views

A probability distribution in n dimensional space which its projection on any line is a uniform distribution?

Does there exist, for any natural $n$, a probability distribution in $\mathbb{R}^n$ whose projection on any line is a uniform distribution?
-3
votes
0answers
17 views

Radial distribution and asset distribution [closed]

What are the classes of radial distribution that can describe asset value/returns distribution?
0
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0answers
72 views

Why is this distribution exponential?

Take the interval $[0, 1]$. Now sample 10000 points in this interval randomly according to the uniform distribution. The fact is that the distribution of the distances between adjacent points on ...
2
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0answers
21 views

Terminology for research on distributions of inner products

Consider a set of vectors $M$ from an inner product space $V$. The ordered set of inner products of all pairs of elements in $M$ uniquely characterizes $M$ up to isomorphism. Suppose now that $V$ is ...
7
votes
7answers
547 views

Semicircle law universality elsewhere

Wigner's semicircle distribution is: $$f(x)=\frac{1}{2 \pi}\sqrt{4-x^2}, \ \ -2\leq x\leq 2.$$ Under reasonable conditions, the rescaled eigenvalue density of random symmetric matrices $M_n$ follows ...
-1
votes
0answers
14 views

Calculate the maximum probability of a result of rolling n dice of varying number of faces [migrated]

Disclaimer: I'm a computer programmer more than a mathematician, so reading text like that of the answer to this question is a little (read: a lot) over my head. I've written an algorithm ...
1
vote
0answers
20 views

Specifying Skellam parameters by given probabilities

The problem sounds quite easy and I still think it is. I somehow have the feeling that I just went too far and just miss the easiest solution now. The numerical solution I came up with is just not ...
0
votes
0answers
17 views

Probability distribution of a Discrete Time Markov Chain between specified start/end states and times?

You've got a DTMC $X(t)$, with state space $0\leq x \leq M$, and transition matrix $Q$. It has absorbing states $0$ and $M$ and otherwise can transition freely (no entries in $Q$ are $0$ other than ...
2
votes
2answers
127 views

Are all mixtures of these unimodal functions unimodal?

Let us say that a function $F\colon(0,\infty)\to\mathbb{R}$ is increasing-decreasing if, for some $c\in[0,\infty]$, $F$ is non-decreasing on $(0,c]$ and non-increasing on $[c,\infty)$. Is it true that ...
-1
votes
0answers
17 views

Sample discrete multivariate normal

I have asked this quesiton on the stats stack exchange with no response, so I'm posting here in hopes that someone may be able to help. What is an efficient way of sampling from a discrete ...
0
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0answers
70 views

A question concerning distribution of $\mathbf{Y}/\|\mathbf{Y}\|_2$ where $\mathbf{Y}\sim \mathcal{N}(\boldsymbol{\mu},\mathbf{I})$

I know that when $\mathbf{Y}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$, $\mathbf{Y}/\|\mathbf{Y}\|_2$ is distributed uniformly on the unit sphere. But to my surprise, I failed to find a simple closed ...
0
votes
0answers
63 views

Probabilities in a directed graph

Given a directed graph of "n" vertices, having on average "m" out-edges each, what is the probability that an arbitrarily chosen vertex will belong to a unique circuit? Also, how does that ...
0
votes
0answers
55 views

Distribution of dot product of two unit random vectors

Consider $\mathbf{u}, \mathbf{v}\in \mathcal{C}^M$ to be two independent unit norm random vectors on the $M-1$ dimensional complex sphere $\mathcal{S}^{M-1}$. In addition, $\mathbf{u}$ follows an ...
0
votes
0answers
114 views

Finitely additive measure over integers [duplicate]

We know that, with Axiom of Choice (AC), it can be shown that there exists a finitely additive uniform distribution defined for all subsets of the integers (see, e.g., Hrbacek and Jech 1999, Ch. 11). ...
1
vote
0answers
38 views

Lower bound for the probability that a certain component of a Gaussian vector dominates all others

Let $X\sim\cal N(\mu,\Sigma)$ be an $n$-dimensional Gaussian vector. I would like to estimate $$P(X_1>\max_{k=2,\dots,n}X_k).$$ While no closed form solution exists (see e.g. MO question on ...
0
votes
1answer
49 views

Finding the distribution of a random variable numerically with sample data? [closed]

Just a thought that I had recently. Suppose given discrete data points for a random variable, could one numerically generate the probability function values at these discrete values? I tried looking ...
1
vote
1answer
168 views

Is regularity closed under products?

Let $G \colon [0,1] \to [0,1]$ be a differentiable cumulative distribution function (monotonically non-decreasing function with $G(0) = 0$ and $G(1) = 1$). We say that $G$ is regular if $$ x - ...
0
votes
2answers
176 views

How do I Calculate :$\int_{0}^{1}x^{k}\psi(x)dx$ where $k\geq 3$ is an integer?

How do I Calculate, if possible, in terms of well-known constants the integral : $\int_{0}^{1}x^{k}\psi(x)dx$ , where $k\geq 3$ is an integer ? note: $\psi(x)$ is digamma function. Any help would ...
2
votes
2answers
260 views

Suggestions for dealing with the “timed” balls-into-bins model

Definitions: Let $T$ (for "time") be a random variable $T \sim \text{Exp}(\lambda)$ and $\Delta t$ is a realization (or called an observed value) of $T$. Let $D$ (for "delay") be a random variable $D ...
2
votes
0answers
35 views

Writing a function as a sum of functions of bounded diameter

This problem is distilled from one arising in a study of complex random variables, but I've removed as much baggage as I can without (I hope) making it trivial. Fix $D>0$. A function $f:\mathbb ...
1
vote
0answers
24 views

Is there an equivalent form for Wishart to a power times a normal?

Lin described two equivalent characterizations of the multivariate t-distribution, viz. As a normal vector divided by an independent chi. That is, $t = Z / \sqrt{\chi^2/v}$, where $Z$ is ...
0
votes
1answer
51 views

How can two random variables are continuous infers that their jointly random variable is continuous [closed]

We assume that $\forall a,b$ suchthat $a^2+b^2>0$, $aX+bY$ is continuous random variable. But we don't assume that $X$ and $Y$ are independent. My question is the following: Is it true that the ...
2
votes
1answer
118 views

What is the distribution of the maximum nearest-neighbor distance of a point cloud sampled from a solid body like?

Let $\mathcal{B} \subseteq \mathbb{R}^n$ be an $n$-dimensional solid body. Assume that we sample $N$ points, say $S = \{ x_1, ..., x_N \}$, from $\mathcal{B}$ uniformly at random. Consider the ...
2
votes
0answers
40 views

expectation involving normal pdf and Rayleigh distribution

I need to calculate following definite integral \begin{equation*} \frac{1}{2\pi }\int_0^\infty \frac{x^2 e^{-x^2/\sigma^2 } }{\sigma} \frac{e^{-\frac{\lambda}{{ax^2+b}}}}{\sqrt{ax^2+b}} ~~dx. ...
2
votes
0answers
51 views

Compute the smoothing of functions

Given a function $g:R^d\rightarrow R$, which is not necessarily continuous, I want to compute the "smoothing" of $g$, i.e., $G(\vec{y})=\int_{R^n} g(\vec{x}) f_{\vec{y}, \sigma}(\vec{x}) d\vec{x} $ ...
4
votes
2answers
286 views

Expectation of Mahalanobis norm

Let $(g_i)_{i=1,...,d}$ sampled i.i.d. from a standard Gaussian, and $(\lambda_i)_{i=1,...,d}$ non-random s.t. $\max_i(\lambda_i)=1$ and $\lambda_i>0, \forall i$. I am looking for the expectation ...
7
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2answers
223 views

A moment problem

Suppose $X, Y$ are two positive random variables such that $\mathbb{E}[X^\alpha] = \mathbb{E}[Y^\alpha]$ for all $\alpha \in (0, 1/2)$. It is also known that the first moment exists for each of them, ...
2
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0answers
78 views

Implication of MGF inequality

Let X and Y be two random variables. Denote by $F_X(x)$ and $F_Y(y)$ their CDFs and by $M_X(t)$ and $M_Y(t)$ their MGFs. It is known that X and Y have the same CDF iff they have the same MGF. My ...
5
votes
0answers
120 views

Elementary function relative to erf

The modified Bessel function of the 1st kind $I_0$ is defined by $$ I_0(z)=\frac1\pi\int_0^{2\pi}e^{z\cos\theta}\,d\theta $$ and arises, among other places, in the probability density function of a ...
2
votes
1answer
101 views

Variant of Skorokhod's theorem

Consider the following situation: $S, T$ are standard Borel spaces (say $S = [0,1]^k$, $T = [0,1]$ if it is helpful). There is a a random variable $\zeta: \Omega \to S$. $f_n(\zeta) \to^d \eta$, ...
2
votes
0answers
35 views

logconcave distribution $f(t)$ leads to concave moments $\mu(x)$. logconvex distribution $f(t)$ leads to convex moments $\nu(x)$?

Let $$\mu_x=\frac{1}{\Gamma(x+1)}\int_0^{\infty}u^x f(u) du \tag{1}$$ Suppose that $f(u)>0$ when $u>0$ and $f(u)\to 0$ fast enough when $u\to\infty$ so that $\mu_x,-1<x<\infty$ ...
1
vote
1answer
180 views

About expectation norms on graphs

Let $S \subseteq V$ of a $d-$regular graph $G$ such that $\mu = \frac{\vert S \vert }{\vert V \vert } $. Let $A$ be the adjacency matrix of the graph. Then define the quantity $\phi(S)= ...
5
votes
2answers
109 views

Finding joint probability from double marginals

Consider three probability distributions in the form $p_1(y,z),p_2(x,z),p_3(x,y)$. When does a global joint probability $p(x,y,z)$ (possibly not unique) exist? The first compatibility condition to ...
4
votes
2answers
166 views

Probable direction of deviations from the expected value in binomial and hypergeometric cases

Suppose I have an urn with N marbles, with frequencies p and q for red and black marbles, and with p > 0,5. I take a sample of r marbles. It sounds intuitive to say that deviations from the mean ...
1
vote
1answer
89 views

A differential inequality and a special value

Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq ...
1
vote
2answers
225 views

Variance of truncated normal distribution

Let $ X \sim \mathcal{N} ( \mu, \sigma^2 ) $, $ - \infty \leqslant a < b \leqslant +\infty $ ($ a, b \ne \infty $ simultaneously) and $ Y $ has a truncated normal distribution on $ (a, b )$, i.e. ...
2
votes
2answers
147 views

Is this a sufficient condition for joint normal distribution?

Suppose I have a random vector $\boldsymbol{Z}$, if I can prove that for $\forall \boldsymbol{\lambda} \neq \boldsymbol{0}$ where $\boldsymbol{\lambda}$ is a fixed vector, not a random vector, ...
1
vote
0answers
31 views

Bound on the total variation distance for multiple samples $d_{tv}(P^n,Q^n)$

Given two discrete distributions $P$ and $Q$, with computable total variation distance $d_{TV}(P,Q)=||P - Q||_1$, is there a precise bound for $d_{TV}(P^n,Q^n)=||P^n - Q^n||_1$, as need to estimate ...
0
votes
1answer
62 views

Name of distribution [closed]

I am searching for the name of the following distribution on the set of positive integers (including zero). Let $C\in \mathbb{Z}_+$ and $n\in \mathbb{N}$ are fixed. Vector $p = (p_1,\ldots,p_n)$ is ...
0
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0answers
117 views

Log-concavity of convolution log-concave and not log concave density functions

Let $Z = X +Y $ be a random variable (r.v.) where $X$ and $Y$ are independent r.vs. If the density function of $X$ and $Y$, $f(x)$ and $g(x)$ are log-concave in the support of $X$ and $Y$, ...
5
votes
1answer
346 views

Square root of normal distribution

Let $X$ and $Y$ be independent random variates with the same probability distribution, $P(x)$. Assuming that the product $Z=XY$ is a random variate with normal distribution, say $$f_Z(x) = ...
1
vote
0answers
62 views

Approximate determinantal point process

Consider a random process defined on $2^{\mathcal{X}}$, i.e. all subsets of a set $\mathcal{X}$. It's well known that this process is determinantal if one can find a positive semidefinite matrix K, ...
2
votes
2answers
176 views

Does $X_n \xrightarrow{d} N(0,1)$ and $X_n/Y_n \xrightarrow{d} N(0,1)$ imply that $Y_n \xrightarrow{d} 1$?

I'm thinking about the following question: If $X_n$ and $X_n/Y_n$ both converge in distribution towards a standard Gaussian random variable and $Y_n \geq 0$ for all $n$, does then $Y_n$ necessarily ...
1
vote
0answers
44 views

Maximum likelihood estimation with several distributions

My question concerns using Maximum likelihood to estimate unknown parameters used by several (poisson) distributions. The parameters are the pairs $(a_1,b_1),\dots,(a_N,b_N)$, and for each pair ...
6
votes
1answer
244 views

Reference on (discrete) log-concave probability distributions

A discrete distribution $p$ over $\mathbb{N}$ is said to be log-concave if it satisfies the following conditions: The support of $p$ is a contiguous interval, i.e. $\exists a \leq b$ s.t. $p_i > ...
3
votes
1answer
130 views

Lower bound on the tail of the hypergeometric distribution

Suppose there is a bag with $M$ white marbles and $N - M$ black marbles. Let $H(n, N, M)$ be a random variable which is number of white marbles in a draw, without replacement, of $n$ marbles from a ...
0
votes
1answer
161 views

Volume of randomly changing sphere follows beta distribution

We are given $X,X_1,\ldots,X_N$ independent and identically distributed $k$-dimensional vectors. For a given query point $X_q\in\mathbb{R}^k$ assume without loss of generality that $X_1,\ldots,X_m$ ...