Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
9,022 questions
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Finding a distribution family that is preserved under mixture.
Consider the following
$f_{t+1}(z)=p_{12} f_{t}(z/A)+ p_{21} f_{t}(z/B)+p_{22} f_{t}(z/(A+B))$, where $A$, $B$, and the $p$'s are constants and $f_t$ is a probability distribution. Are there any nice ...
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Statistical test comparing two relative frequencies
I'm working with four populations consisting of true/false events. They each have a different mean and variance. I have samples from each, with different sample sizes. Call the percentage of observed ...
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Probabilities and rolling 2 dice
Suppose you start at position 0. You then roll 2 6-sided dice. You move to the integer, call it z, that is the sum of the two dice. You then roll again. If the result of the roll is z', you move ...
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What's the expected number of iterations for this process?
Each step of the process consists of choosing a random integer between 1 and the last number chosen this way. On average, how long does it take to obtain "1" as a result of this process for any given ...
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Extreme value theory
I have been asked to provide an "approximation at infinity" of an expression that at the end simplifies to $-\frac{b e^{-a t}-a e^{-b t}}{a-b}$, in a course about extreme value theory.
In the course, ...
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How to show that an infinite sequence is normal if and only if every block of equal length appears with equal frequency?
An infinite sequence is normal if all strings of equal length occur with equal asymptotic frequency.
Formally, let $\Sigma$ be a finite alphabet of $b$ digits. Let $S$ be an infinite sequence and $\...
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A geometric interpretation of independence?
Consider the set of random variables with zero mean and finite second moment. This is a vector space, and $\langle X, Y \rangle = E[XY]$ is a valid inner product on it. Uncorrelated random variables ...
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When does a probability measure take all values in the unit interval?
Let $\mathbb{P}$ be a probability measure on some probability space $(\Omega,\mathcal{A})$. Are there conditions on the $\sigma$-algebra $\mathcal{A}$ such that for every real number $c\in [0,1]$ we ...
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Distribution of the sum of the $m$ smallest values in a sample of size $n$
Let $\mathbf X = [X_1, X_2, \ldots, X_n]^\mbox{T}$ be a vector random variable drawn from a known distribution with CDF $F(x)$. The CDF for the minimum value in $\mathbf X$ is clearly $P[\min_{i=1\...
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Binomial distribution parity
Let $X \text{~} \text{Binomial}(n, p)$.
What is $\text{P}[X \mod 2 = 0]$? Is it of the form $1/2 + O(1/2^n)$?
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Probability of n k-sided dice showing exactly m different faces
I found the following closed form solution for the abovementioned problem:
$${1\over k^n}\cdot{k!\over (k-m)!}\cdot{\{{n\over m}\}}$$ with ${\{{n\over m}\}}$ being the Stirling Number of the second ...
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MicroArray, tesing if a sample is the same with high variance data.
I'll explain the problem but what I am looking for is a few suggested methods to approach this problem.
You don't need to know what a microarray but if you are interested look here link text
The info ...
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limsup and liminf for a sequence of sets
how does limsup and liminf for a sequence of sets, apply to probability theory. any real world examples would be much appreciated
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Hitting times for an N-dimensional random walk on a lattice with (strictly positive) random integer steps
Please consider a random walk on a finite N-dimensional lattice with vectors $(x_1, ..., x_N)$. We define the origin to be $(0, ..., 0)$ and the target to be at the point in the lattice furthest away ...
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What m minimizes E(|m-X|^3) for a random variable X?
Let X be a random variable. Then E(|m-X|^1) is minimized when (as a function of m) when m is the median of X, and E(|m-X|^2) is minimized when m is the mean of x.
A couple weeks ago in a technical ...
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Has the Lie group preserving a probability distribution been used in Bayesian statistics?
For a (possibly signed) nondegenerate probability measure $\pi$ on $\{1,\dots,n\}$ define
$$\langle \pi \rangle := \{R \in \operatorname{STO}(n): \pi R = \pi \}.$$
Here $\operatorname{STO}(n)$ denotes ...
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Joint Law with 2 marginals and marginal of the spread
I have a question for you and thank you in advance for your answers and ideas.
Let us suppose that we have the marginal distributions of two r.v X and Y, and also the law of X-Y (or any linear ...
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randomness in nature [closed]
What is the explanation of the apparent randomness of high-level phenomena in nature?
For example the distribution of females vs. males in a population (I am referring to randomness in terms of the ...
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An Easy Sanov-Type Theorem for Markov Chains?
First, the (simple!) setup:
I have a Markov chain X t on some finite state space Ω with stationary distribution π, and a function f from Ω to R. I'd like to estimate the integral of ...
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Problem with Pearson correlation coefficient. [closed]
I have two random variables X and Y. X follows a power law distribution. I know its generating function G(x). I also know the Pearson correlation coefficient of X and Y. How do I find the generating ...
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Is this a well-known probabilistic model?
While I was thinking about the Erdős discrepancy problem, the following random walk model arose rather naturally. You fix a positive integer k, and you take a random step of 1 or -1 at each stage,...
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Non-existence of integral with respect to Poisson Random Measure
Let $\xi$ be a Poisson Random Measure of intensity $\mu$ (informally $\mathbb E\xi = \mu$).
(For $f \ge 0$, say) when does $\xi f = \infty?$ Kallenberg (Foundations of Modern Probabilility) claims ...
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A formula for moments of the limit distribution of singular values in the proof of the circular law
One of the steps in the proof of the circular law in random matrix theory is obtaining the limiting spectral distribution for the matrix
$(\frac{1}{\sqrt{n}} X_n - zI)(\frac{1}{\sqrt{n}} X_n - zI)^\...
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Equality in the union bound.
Lemma: Let $A_1,\ldots,A_n$ are events $n\in\mathbb{N}$ then
$$
\sum_{i=1}^n \mathbb{P}(A_i) = \mathbb{P}(\cup_{i=1}^n A_i)
$$
if and only if $A_1,\ldots,A_n$ are mutually exclusive.
Both ways are ...
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How many trial picks expectedly sufficient to cover a sample space?
Consider a sequence of independent events where an $r$ element subset of an $n$ element set is picked uniformly randomly (ie. any of the $\begin{pmatrix}n\newline r\end{pmatrix}$ possibilities being ...
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Average distance between numbers of the form $2^{a}3^{b}$
I want to order all numbers of the form $2^a3^b$. I need to find the average distance between a random consecutive pair.
For example, in case of a random consecutive pair $2^{n'}$ and $2^{n'+1}$, the ...
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Sample from uniform distribution vs. Sample from random distribution
I could sample a set of m elements from the uniform distribution over a universe $U$ of n >> m elements. Alternately, I could select a random probability distribution $\mathcal{D}$, and sample $m$ ...
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Where can I learn about master equation?
I am reading a paper by Dorogovstev on structure of growing complex networks with preferential linking. I need to learn master equation for this.
I need a reference for the same.
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Number of uniform rvs needed to cross a threshold
Let $N(x)$ be the number of uniform random variables (distributed in $[0,1]$) that one needs to add for the sum to cross $x$ ($x > 0$). The expected value of $N(x)$ can be calculated and it is a ...
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Integral determines function behaviour
Let us define:
$f(t) = t^{-1} \int_{\mathbf{R}^{3}} Exp[-\frac{x^2}{2t}] h(x) dx,$
for a real function h. What can I say about this function if I know that
$f(t) \rightarrow 1$.
I think that the ...
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A random variable: is it a function or an equivalence class of functions? [closed]
A random variable: is it a function or an equivalence class of functions?
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Correlation of Statistical Tests
Suppose I have a sequence $\{x_i\}_{i=1}^\infty$ of zeros and ones. I want to test if they are randomly generated according to a conjectured scheme (the example to keep in mind is that they are ...
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Extension of some feature of SDE Ornstein-Uhlenbeck type
Hi everyone,
I am looking for some ideas (or references) in order to get an explicit SDE (if it exists) which would have a stylised property extending in some sense the mean-reversion property of SDE ...
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Local view of setting p*n out of n bits to 1
For p a constant in (0,1) and n going to infinity such that pn is an integer,
consider the distribution on n bits that selects a random subset of pn bits, sets those to 1, and sets the others to 0.
...
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Intuitive explanation to Probability question [closed]
I have \$3. I flip a coin. If I get heads, I get \$1. If I get tails, I lose \$1. The game stops when I have \$0 or \$7. What is the probability I get \$7?
I solved this by creating a system of ...
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Name for probabilistic version of Pascal's identity and differentiation formula for binomial distribution
I'm trying to find a standard name or standard reference for two simple-to-prove relations involving binomial distributions.
Define:
$b(n,r,p) := \binom{n}{r}p^r(1 - p)^{n-r}$
i.e., it is the ...
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How does the Dirichlet process work?
Hi, i'm looking to get into nonparametric bayesian techniques but I'm having problem understanding what's going on in the definition of the Dirichlet process or how it works. So what does P ~ DP(&...
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Is there any random variable which has unbounded fourth moment? [closed]
In many statements in probability, there is an assumption like bounded fourth moment. So is there any random variable which has unbounded fourth moment?
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Counting lattice points on an n-simplex
Imagine an n-simplex, the solution set for the expression: $a_1$*$x_1$ + $a_2$*$x_2$ + ... + $a_n$*$x_n$ = S, where:
$a_1$ through $a_n$ are positive bounded integers
$x_1$ through $x_n$ are ...
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Efficiently sampling points from an integer lattice.
Let $\mathcal{L}$ = {x$\in$ $N^n$ : ||x||$_1$ $\leq$ m} denote the set of integer points in the positive orthant of the $\ell_1$ ball of radius $m$, where $m < n$. For each $x \in \mathcal{L}$, let ...
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'Focusing' the mass of the Probability Density Function for a Random Walk
Consider a random walk on a two-dimensional surface with circular reflecting boundary conditions (say, of radius 'R'). Here, for a fixed-size area, one finds a larger fraction of the probability ...
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When does a pointwise CLT hold?
Let $X$ be a random variable with mean $0$ and variance $1$, and let $X_1, X_2, X_3, \dots$ be iid copies of $X$. Under what conditions can we say that the density of $\frac{X_1+\dots+X_n}{\sqrt{n}}$ ...
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easy(?) probability/diff eq. question
I've been wondering about this ever since I was a little kid and I used to ride in the back of the car and my mom would speed like hell towards a green light, only to slam on the brakes when she ...
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Difference between Beta Process and Dirichlet process
I'm trying to understand the definition of a Beta process, as given in the paper:
www.ece.duke.edu/~lcarin/Paisley_BP-FA_ICML.pdf
The problem is that from the definition it follows that every ...
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Probability distribution of the median
Suppose we have $2k + 1$ points $a_1, ..., a_{2k+1}$. Each point is uniformly distributed between 0 and N. What is the distribution of the median (i.e. of the k+1-th point) ?
What happens if $a_1, ......
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Concentration of measure for gaussian inner products
There exists extensive theory for the concentration of Gaussian measure. Through that, it can be easily proved that the square of the $\ell_2$ norm of a length $n$ zero mean Gaussian vector ${\bf x}$ ...
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Range of binomial probability, given a certain number of observations?
Let's say I am given $n$ flips of a coin, $k$ of which are heads.
These are iid flips.
Can I say, with probability $p > 1/2$, that the true probability of heads is in range $[p_1, p_2]$ ?
What is ...
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Random values and their probability of reoccuring [closed]
I have a web application that prompts users to answer a question when the computer they are using is not recognized. A user complained today saying she is always prompted for the same question. I ...
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for a natural exponential family, A is the cumulant function of h?
Reading "Monte Carlo Statistical Methods" by Robert and Casella, they mention that if
$f(x) = h(x) \exp(\langle \theta, x \rangle - A(\theta))$
defines a family of distributions for $X$, parametrized ...
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Probability vertices are adjacent in a polygon
With regard to my original question:
A subset of k vertices is chosen from the vertices of a regular N-gon. What is the probability that two vertices are adjacent?
I suppose that the responses ...
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