All Questions
Tagged with pr.probability probability-distributions
1,384 questions
12
votes
3
answers
2k
views
How to efficiently sample uniformly from the set of $p$-partitions of an $n$-set?
Let $n,p \in \mathbb{N}_+$ with $p \leq n.$ Let $\mathcal{P}$ denote the set of partitions of $\{1, \ldots, n\}$ into $p$ nonempty sets. How can I efficiently sample uniformly from $\mathcal{P}$?
- 922
13
votes
1
answer
3k
views
What is the maximum-entropy distribution given mean, variance, skewness, and kurtosis?
$X\in \mathbb{R}$. Which distribution $P(X)$ has the highest possible entropy given its expected value, variance, skewness, and kurtosis? Is it an exponential family distribution of the form $P(X) \...
- 233
5
votes
2
answers
683
views
Asymptotic Expansion of Distribution in Central Limit Theorem for Non-Identically Distributed Random Variables
My question is related to the following theorem (e.g. Section XVI.4 of Feller's 1971 book): Let $Z_i$ $(i=1,\cdots,n)$ be independent and identically distributed random variables with mean zero, ...
- 325
0
votes
1
answer
2k
views
Markov Chain: state reduction
Hi I am trying to understand a proof in a paper (written by Isaac Sonin), I don't know if anyone could give me a clarification on the following:
Firstly we have a Markov chain $\{Y_k\}$ with finite ...
- 23
3
votes
2
answers
566
views
Moments of Matrix Gamma distribution
Matrix gamma distribution (defined for example in http://en.wikipedia.org/wiki/Matrix_gamma_distribution) is one way to generalize Wishart distribution. In our course work that distribution was used ...
- 31
1
vote
0
answers
190
views
Nontransitive dice: the least number of faces?
Here is an introduction to nontransitive dice. The question is: given $n$-player with a $m$-sided dice each one, the what is the minimum of $m$ for a fixed $n$ to produce nontransitivity?
Here is ...
- 273
2
votes
1
answer
354
views
star-product of copulas
I have recently come accross the star product of copulas, that is if $A$ and $B$ are 2-copulas and $\{C_t\}_{t\in[0,1]}$ is a family of copulas, then $C(x,y,z) = \int_0^y C_t(\frac{\partial}{\partial ...
- 23
13
votes
1
answer
1k
views
Does $P(X_1>X_2)$ and $P(X_1=X_2)$, where $X_1$ and $X_2$ are independent and Poisson distributed, uniquely determine the parameters?
Let $X_1$ and $X_2$ be independent Poisson distributed random variables with parameters $\lambda_1$ and $\lambda_2$, respectively.
Let $a = P(X_1 > X_2)$ and $b = P(X_1 = X_2)$.
Question: ...
- 326
2
votes
1
answer
231
views
First colour to be drawn $n$ times from a hypergeometric distribution
Consider a set up of drawing balls without replacement from a population of $R$ red balls and $B$ black balls. Is anyone aware of an efficient way to calculate the probability, $p$, that I draw $n$ ...
- 121
5
votes
1
answer
765
views
Measure concentration for weakly dependent random variables
For an application quite alien to probability theory, I'd like to have a kind of measure concentration estimate, in the following spirit. Suppose that to every $1\le i,j\le n$ there corresponds a zero-...
- 23k
1
vote
1
answer
478
views
Distance between the product of marginal distributions and the joint distribution
Given a joint distribution $P(A,B,C)$, we can compute various marginal distributions. Now suppose:
\begin{align}
P1(A,B,C) &= P(A) P(B) P(C) \\
P2(A,B,C) &= P(A,B) P(C) \\
P3(A,B,C) &= P(...
- 49
0
votes
2
answers
174
views
Joint distribution with specified marginals
Suppose we are given a probability distribution over a finite discrete product space $p(x,y)$ with marginals $p(x), p(y) > 0$ for each $x,y$ respectively. We are given two more marginal ...
- 1,269
16
votes
6
answers
3k
views
A normal distribution inequality
Let $n(x) := \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) := \int_{-\infty}^x n(t)dt$. I have plotted the curves of the both sides of the following inequality. The graph shows that the ...
- 2,239
1
vote
1
answer
228
views
Is anything known about Large Deviation Principle for non additive functionals on Markov chains?
Let $\Sigma$ be a finite set of cardinality $|\Sigma |$ and
$$\Pi = \{ \pi(i,j)\}_{i,j = 1}^{|\Sigma|}$$
a stochastic matrix (ie a matrix whose elements are non negative and such that
each row sum ...
- 3,245
11
votes
1
answer
2k
views
Generalized central limit theorem
I am looking for a generalized central limit theorem for non-square integrable stationary sequences. More precisely I suspect that when $(X_j)_{j\geqslant 1}$ is a stationary sequence such that $X_i$ ...
- 1,491
7
votes
1
answer
12k
views
inner product of two gaussian random vectors?
Suppose that $x, y\sim N(0,I_n)$ are independent. Consider the inner product $\langle x, y\rangle$. Intuitively, $y$ behaves like a random vector of length $\sqrt n$, so $\langle x, y\rangle$ is close ...
- 145
2
votes
0
answers
341
views
Marginalizing multivariate normal over defined interval
Hello everyone,
I am trying to obtain an analytic expression for the following Gaussian integral
$$\frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \int \kern-0.2em \cdots \kern-0.2em \int d\mathbf{x}_{\sim i} \;...
- 21
4
votes
2
answers
1k
views
Probability distribution over cluster size in Erdős–Rényi random graph.
My question is about the probability distribution over the possible size of the containing cluster of a randomly chosen node in an Erdős–Rényi random graph.
Let G(n,p) be an Erdős–Rényi random graph (...
- 41
7
votes
1
answer
3k
views
What is characteristic function of maximum of i.i.d. random variables?
Is is possible to get characteristic function of maximum of i.i.d. random variable sequence? Such as $X_1, X_2$ are two i.i.d random variables, then what is characteristic function of $X=\max(X_1,X_2)$...
- 83
1
vote
0
answers
442
views
Joint distribution from multiple marginals
Consider an experiment consisting of a repeated trial with two random Bernoulli (=binary) variables, A and B. Each trial consists of multiple outcomes for both A and B. Each trial has the same number ...
- 113
1
vote
3
answers
314
views
A sampling and learning question
Suppose there is an oracle that returns a number $b \in \mathbb{Z}_{n}$ whenever I press the button.
We have $b = a + e$, where $a \in \mathbb{Z}_n$ is a fixed number and $e$ is sampled according to ...
- 11
5
votes
0
answers
96
views
Is there a name for the set of distributions whose probability generating functions are Mobius transformations?
Consider a discrete random variable $N\in\mathbb N$ with
$\mathbb P(N=0) = p$,
$\mathbb P(N=n) = (1-p)(1-q)q^n$ for $n\neq 0$.
Then the probability generating function of $N$
$$\mathbb E(z^N) = \...
- 241
3
votes
0
answers
104
views
Minimizing/Maximizing the tail of the convex combinations of Chi Squared i.i.d random variables
Consider $N$ i.i.d random variables, $X_{1}, X_{2}, \ldots, X_{N}$ , that are chi-squared of degree $K \geq 2$. Also consider the following 3 vectors:
\begin{eqnarray*}
\bar{a} &=& (\frac{1}{...
- 51
4
votes
1
answer
700
views
Bounding statistical distance by matching moments
Suppose we have distributions $p(x)$ and $q(x)$ both supported on integers in $[-n, +n]$. We want $p$ and $q$ to have statistical (total variational) distance of at most $\epsilon$.
Is there a ...
- 1,324
0
votes
1
answer
1k
views
Expected value with a kronecker product and Gaussian distributional assumption
What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $? The kronecker product is where the confusion ...
- 101
1
vote
0
answers
1k
views
Expected value of the log of the factorial of a poisson distribution
I found the expression for the expected value of the falling factorial of a Poisson distribution ($\lambda^n$) from - http://en.wikipedia.org/wiki/Factorial_moment. Is there a similar expression for ...
- 171
4
votes
3
answers
910
views
Solution to the fractional differential equation
What is the solution of the fractional differential equation
$$
f^{(\alpha-1)}(t) = tf(t)
$$
where $(\alpha)$ denotes the fractional derivative of order $\alpha$
EDIT: Background behind this ...
- 329
2
votes
0
answers
198
views
Have you seen this one parameter family of distributions before?
This is a one parameter family of distributions. Choose some parameter $\lambda > 0$ and define the measure $\nu_\lambda$ which is absolutly continuous with respect to the Lebsegue measure with the ...
- 21
2
votes
2
answers
869
views
Computing hypergeometric function of matrix argument
In the context of the Bingham probability distribution the ${ }_1F_1$ hypergeometric function of matrix argument naturally arises as a normalization constant of the probability distribution function. ...
- 73
0
votes
1
answer
229
views
Weak convergence in measure for negligible sets.
Let $X$ be a Polish space and $(P_n)$ a sequence of Borel probabilities which converges weakly in measure to a Borel probability $P$. By this i mean that for any $f\in C_b(X)$ which is continuous and ...
- 125
2
votes
0
answers
141
views
question about circular law
Hi,
I have a question about the circular law.
Consider $A_n=[x_{ij}]$ a sequence of random matrices where $x_{ij}$ are iid with mean $0$ and variance $1$. Consider $\lambda_{n,1},\dots,\lambda_{n,n}$ ...
- 29
4
votes
3
answers
433
views
Expectation of $(c+e^{N(0,\sigma^2)})^{-n},\, n>0$
I would like to know if there's a way to compute or approximate the following expectation:
$$\mathbb{E}[(c+e^X)^{-n}]$$
where $X=N(0,\sigma^2)$ and $n,c>0$ (you can also assume that $n$ is a ...
- 81
2
votes
1
answer
576
views
Inequality for square of the subgaussian distributions
Hi all,
For my research I am trying to bound some exponential moments of subgaussian r.v.'s. And I am stuck with proving one of such inequalities. More specifically:
Let $a$ be unit vector in $\...
- 51
3
votes
1
answer
354
views
Central Limit Theorem for additive function of permutations of sequences
Fix the finite sets $\mathcal{X}$ and $\mathcal{Y}$, and probability mass functions $P_X(x)$ and $P_Y(y)$ on these sets. Assume each value of $P_X(x)$ and $P_Y(y)$ is rational.
For each $n$ such ...
- 325
5
votes
1
answer
778
views
Calculate channel capacity of general channel under constraint
Given a conditional distribution $P_{Y\mid X}$ I'd like to find the prior distribution $P_X$ that maximizes the mutual information $I(X;Y)$ with $P_Y(y)=\int P_{Y\mid X}(y\mid x)P_X(x) \, \text{d}x$ (...
- 59
0
votes
2
answers
298
views
Are all variables in a set of random variables independent if all pairs are independent?
If I have a sequence of random variables $X_1, X_2, \ldots, X_n$ (possibly infinite) such that all pairwise cdf's are factorized:
$$F(X_i, X_j) = F_i(X_i) F_j(X_j)$$
for all pairs $(X_i, X_j)$, does ...
- 923
2
votes
1
answer
2k
views
Is an L_1 bounded sequence of random variables with uniformly converging CDFs uniformly integrable?
Changing my question in light of Dan's answer. Thanks, Dan.
Consider a sequence of real random variables $X_i$ bounded in $L_1$, that is $\mathbb E\left|X_i\right|\leq M$ for all $i$. Suppose that ...
- 23
14
votes
3
answers
2k
views
Concentration bounds for sums of random variables of permutations
I'm trying to find theorems regarding random variables derived from sampling permutations, specifically concentration bounds.
As an example, let $X_i$ be the $\{0,1\}$-random variable that represents ...
- 539
7
votes
1
answer
346
views
Probability density that minimizes the sample range
Let $\mathcal{F}$ denote the set of all "concave probability distributions" on the unit interval, that is, all functions $f:[0,1]\to \mathbb{R}$ such that $f$ is concave, $f(x)\geq 0$ for all $x\in [0,...
- 73
9
votes
2
answers
2k
views
What is the most extreme set 4 or 5 nontransitive n-sided dice?
A set of nontransitive dice is a set of dice whose face numbers are such that the relation "is more likely to roll a higher number than" is not transitive. (See wikipedia)
For some sets, the ...
- 93
0
votes
1
answer
121
views
Multinomial -- how many trials in order to see all the values with prob 1-\alpha
Let suppose that I have a box with $k$ different balls, each one with a different color.
At each time I have to extract a ball and observe the color. Then I put the ball back in the box.
How many ...
- 165
3
votes
3
answers
379
views
Support of an infinitely divisible measure.
Hello,
if $G$ is a compact Lie group. Let $\mu$ be an infinitely divisible measure on $G$, such that $e$, the neutral element of $G$, is in the support of $\mu$. Is that true that the support of $\...
- 31
1
vote
2
answers
4k
views
minimum of different independent Poisson random variables
Let $X_1,\ldots,X_N$ be independent Poisson distributed random variables with unequal parameters $\lambda_1,\ldots,\lambda_N$.
Is there any closed form expression or at least a good approximation for ...
- 171
15
votes
2
answers
6k
views
Distribution of inverse of a random matrix
I got stuck into a problem and couldn't find its
satisfactory answer anywhere.
My question is simple. Suppose I have a fat random matrix (i,e., $R$ has dimensions $k\times d$ where $k<d$) whose
...
- 151
1
vote
1
answer
140
views
Equivalence between choosing a subspace and choosing its orthogonal
Hi,
We consider subspaces of $\mathbb{R}^N$.
Suppose that we have a property called $\mbox{Prop}$ that apply to subspaces of $\mathbb{R}^N$. That is to say a function from the set of subspaces of $\...
- 13
0
votes
1
answer
320
views
Simple markov chain problem
I know this is an easy problem, but I can't figure it out.
A particle takes discrete steps $σ_1,σ_2,σ_3,…,σ_n$ which take on values +1 or −1. However, $P(σ_i=+1)=p$ and $P(σ_i=−1)$ will be $1-p$.
...
- 39
6
votes
1
answer
333
views
Trasportation metric (AKA Earth-Mover's, Wasserstein, etc.) as "natural" / "induced"?
Context: Given a discrete finite metric space $X$ (in my case X={0,1}$^n$ with the Hamming/L$_1$ distance), I need to define the natural or canonical metric on the set of all probability distributions ...
- 111
0
votes
1
answer
333
views
Limit of the stochastic process at time 0
This is not a homework question so please be kind not to remove it right away. I am working on some research but have to justify the following argument: Assume $S_t$ is a continuous stochastic process,...
- 153
0
votes
0
answers
2k
views
Derivative of the most probable value (of a gaussian variable) VS most probable value of the derivative
Let $x$ be a random variable with gaussian probability distribution $P(x)$. We assume that $x$ depends parametrically on a parameter $t$ so that :
$P(x(t))=\frac{1}{\sqrt{2\pi\sigma^2(t)}}\exp(-\frac{(...
- 1
2
votes
1
answer
469
views
If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?
If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
- 3,245