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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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263 views

Sparse representation of a distribution with independent and correlated variables

Here's what I'm trying to do: Imagine a probability distribution over $\mathbf{R}^2$, $P(x,y)$. I can approximate $P(x,y)$ with set of $N$ points $\{(x,y)_i\}$ drawn from $P$. By approximate, I mean ...
4
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0answers
302 views

When is taking an average (mean) an algebraic operation in the sense of monads?

Taking the average of a sequence of numbers is not an "algebraic" operation, in the following sense. Given sequences $X_1,X_2,\ldots,X_n$ of numbers, one could either take the average of each one, ...
4
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2answers
2k views

finding numbers at k hamming distance

Guys, I have N < 2^n randomly generated n-bit numbers stored in a file the lookup for which is expensive. Given a number Y, I have to search for a number in the file that is at most k hamming dist....
18
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0answers
867 views

On random Dirichlet distributions

Fix a dimension $d\ge2$. Let $Q_d$ denote the positive quadrant of $\mathbb{R}^d$, that is, $Q_d$ is the set of points $\mathbf{x}=(x_i)_i$ in $\mathbb{R}^d$ such that $x_i>0$ for every $i$. For ...
4
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2answers
2k views

low-discrepancy sequences for sampling of distributions

Hi everybody, I am writing of physics simulation that traces charged particles. I need to set up these particles in phase space ( 3 space dimension + 3 momentum dimensions) following distributions. ...
3
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1answer
778 views

Restriction of a linear functional equation to surface of a sphere

Let $f_i : R \rightarrow R$ and $g_j: R \rightarrow R$ be unknown functions, for $i = 1, \cdots, N$ and $j = 1, \cdots, K$. Let $A$ be a $K \times N$ matrix whose columns are unit-length vectors ${\...
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2answers
307 views

when does inner product with fixed vectors determine joint distribution?

Given a random vector $(X_1,X_2)$. If $aX_1 + bX_2$ is Gaussian for all pairs $a,b$, then $(X_1,X_2)$ is jointly normal. More generally, is the following statement true? If $aX_1 + bX_2$ has the same ...
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0answers
140 views

What is the complexity of computing the p-values for a multivariate hypergeometric distribution

Are there efficient approaches to compute p-values of samples drawn from a multivariate hypergeometric distribution? Specifically, how does the complexity grow as a function of the "urn size" and the ...
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1answer
412 views

Approximating expectation [closed]

if we are given a finite number N of points drawn from a probability distribution, expectation can be approximated as a finite sum over these points: E[f]=(1/N)(summation of f(x) over these N points). ...
2
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2answers
816 views

Exist closed forms of the distribution of return time in markov chains?

Hi, I am interested in the distribution of return times in simple random walks on finite graphs. Let $G$ be a connected finite graph with, with two independent random walks. If both random walks ...
2
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0answers
302 views

Computable distribution on [0,1] with C-infinity distribution function

Does anyone know of an easily-describable distribution on $[0,1]$ with a density $p$ (with respect to Lebesgue measure) that satisfies the following properties: $p$ is $C^\infty$ $p(0) = a$, $p(1) = ...
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0answers
782 views

For what sub-$\sigma$-algebra are these two measures equivalent?

In two statistics papers (linked inline below) I have come across two definitions of certain probability measures. I conjecture that for particular choices of the construction that they are ...
3
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3answers
2k views

Statistics of a simple Markov chain

Imagine a two-state Markov chain which hops between the states $\pm 1$ with probability $p<1/2$, so that the autocorrelation function after $k$ steps is $\rho_k = (2p-1)^k$ If I take an ...
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2answers
3k views

Weighted Distribution [closed]

Hello, here's my basic problem - I would appreciate any help. We use a weighted system based on 1 (developing) ,2 (performing) ,3 (leading) to evaluate an employee's rating using various metrics. Each ...
2
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1answer
527 views

What are the origin and applications of this result?

In a course taught by Morris Eaton on multivariate statistics that dealt mostly with the Wishart distribution, I learned this proposition: Suppose $$ M = \begin{bmatrix} A & B \\\\ B^T & C \...
4
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3answers
1k views

distribution of $\{na\}$ when $a$ is irrational number

(by $\{x\}$ I mean the fraction part of the real number $x$) If $a$ is an irrational number and $n$ is a integral number, what is the distribution of $\{na\}$? I'm asking for some continuous function $...
0
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2answers
280 views

Relationship between these two probability mass functions.

If I have two different discrete distributions of random variables X and Y, such that their probability mass functions are related as follows: $P(X=x_i) = \lambda\frac{P (Y=x_i)}{x_i} $ what can ...
1
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2answers
13k views

How to ensure the non-negativity of Kullback-Leibler Divergence KLD Metric (Relative Entropy)?

I’m having some problems in ensuring the non-negativity of KLD! I know that KLD is always positive and I went over the proof. However, it doesn’t seem to work for me. In some cases I’m getting ...
5
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2answers
4k views

What is the expected maximum out of a sample (size N) from a geometric distribution?

Lets say I have a geometric distribution (of the number X of Bernoulli trials needed to get a success) with parameter p (success probability of a trial). Assume I ...
4
votes
1answer
322 views

How to calculate the probability of N normal variable being in increasing order?

Suppose we have $n$ normal variable $X_1,X_2,\dots,X_n$, with corresponding mean $\mu_1,\dots,\mu_n$ and sd $\sigma_1,\dots,\sigma_n$. What is the probability of $X_1 < X_2 < \dots < X_n$, i....
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3answers
492 views

Moments of a distribution - any use for partial or higher moments?

It is usual to use second, third and fourth moments of a distribution to describe certain properties. Do partial moments or moments higher than fourth describe any useful properites of a distribtution....
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1answer
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kalman filter: understanding the mathematical part

i am currently reading the Probabilistic robotics book where the filters are discussed. Such filters as kalman filter or particle filters. Now I can understand one thing while reading about the ...
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1answer
205 views

Completeness of an “infinite mixture of gaussians” representation

Is there a complete "infinite mixture of gaussians representation" for densities? What I mean is, is there, for any reasonably big class of densities $\phi(x)$ I can come up with a function $c(\mu, \...
0
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1answer
196 views

Constraints for different probability measures to have the same expectation.

Take different $D_i \in \mathbb{R} \rightarrow \mathbb{R}$ functions $f_1, f_2$ (i.e. $\exists x : f_1(x) \neq f_2(x)$). We have $E[f_1(x)] = E[f_2(x)]$ Are there conditions that $f_1, f_2$ must ...
10
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7answers
17k views

Resultant probability distribution when taking the cosine of gaussian distributed variable

I am trying to do a measurement uncertainty calculation. I have a gaussian distributed phase angle (theta) with a mean of 0 and standard deviation of 16.6666 micro radians. The variance is the ...
2
votes
1answer
2k views

Density function for a multivariate Bernoulli-like distribution

I'm looking for a distribution to model a vector of $k$ binary random variables, $X_1, \ldots, X_k$. Suppose I have observed that $\sum_i X_i = n$. In this case I do not want to treat them as ...
3
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3answers
3k views

“Square root” of Beta(a,b) distribution

Under what conditions on a and b is there a distribution $f_{a,b}$ such that the product $XY$ of two independent realizations $X$ and $Y$ from $f_{a,b}$ has a Beta(a,b) distribution? A standard ...
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1answer
283 views

Probability distribution needed [closed]

Let me clarify my needs. The PDF must comply to: 1. The mean is always in the shorter tail 2. Should have an inverse function 3. Be defined in the interval [0, 1] 4. Should have a shape parameter that ...
6
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3answers
8k views

Approximation of a Normal Distribution function

I am reviewing and documenting a software application (part of a supply chain system) which implements an approximation of a Normal Distribution function; the original documentation mentions the same/...
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4answers
3k views

Differential Entropy of Random Signal

Prove that the Normal (Gaussian) Distribution with a given Variance $ {\sigma}^{2} $ maximizes the Differential Entropy among all distributions with defined and finite 1st Moment and Variance which ...