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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Are there interesting problems involving arbitrarily long time series of small matrices?
Are there well-known or interesting applied problems (especially of the real-time signal processing sort) where arbitrarily long time series of small (say $d \equiv \dim \le 30$ for a nominal bound, ...
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Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance
Problem: Consider a random walk on the lattice $\mathbb{Z}^2$ where on each iteration a particle either stays at its current location or moves to a neighboring vertex with probability 1/5. We start ...
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A question about Chapter 12 (Vapnik-Chervonenkis Theory) of 'A Probabilistic Theory of Pattern Recognition'
Hi,
Can anyone familiar with the book 'A Probabilistic Theory of Pattern
Recognition' or the theory described help me out?
See quote from chapter 12, 'Vapnik-Chervonenkis Theory', of 'A
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how to sample a conditioned diffusion
there are several reasons why we could be interested in sampling conditioned diffusions:
if we observed a diffusion at discrete time and want to do some kind of inference on the parameters of the ...
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probability of subset sum after rolling dice 4 times [closed]
If we roll 4 dices (fair), what is the probability of "sum of subset" being 5. e.g. 1432,1121, 2344, 2354 have a subset sum of 5. Can you illustrate how to calculate this.
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Stationary Solutions of stochastic differential equations
When does the stationary density of an homogeneous Markov process exist?
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Constructive aspects of Caratheodory's theorem in convex analysis
Let me paraphrase Caratheodory's theorem in a probabilistic setup:
Let $X$ be a real-valued random variable. For $k = 1, \ldots, m$, let $f_k: \mathbb{R} \to \mathbb{R}$ be a continuous function such ...
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tetrahedron edges probability
If 6 numbers are chosen at random, uniformly and independently, from the interval [0,1], what is the probability that they are the lengths of the edges of a tetrahedron?
I wrote some code and ...
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The mathematical theory of Feynman integrals
It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly.
Arguably, they are the most important such tool. Briefly, the question I'd like to ...
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Conditional expectation [closed]
Given E[v|X=x]=g[x] and the pdf of X (f[x]), how to calculate E[v|x>=x0]? The pdf of V or the joint pdf of V,X are unknown. My guess is that this problem has no solution.
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Another question on Øksendal's book
Hi
On page 98 "Stochastic differential equations" of Øksendal, 6th edition,
the author writes that $$\int_{0}^{u}\Big(\int_{0}^{t}\frac{\partial}{\partial t}f(s,t)dR_{s}\Big)dt=\int_{0}^{u}\Big(\...
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weak convergence in infinite dimensional spaces
Weak convergence can be tricky when dealing with infinite dimensional spaces. For example, the usual Levy's continuity theorem does not extend readily to separable Banach spaces.
Consider a (...
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Minimum Hamming Distance Distribution in a Random Subset of Binary Vectors+
Select $K$ random binary vectors $Y_i$ of length $m$ uniformly at random.
Let the following collection of random variables be defined: $X_{i,j}=w(Y_i \oplus Y_j)$ where $w(\cdot)$ denotes the Hamming ...
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Mathematical means of studying large and complex but finite systems?
I want a list of the sort of mathematics/mathematical tools that are applied to the study of complex and probabilistic systems in order to make quantitative and qualitative observations about their ...
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Comparing normally distributed variables
Given two normally distributed variables x_1, x_2, is there a non-simulation method of calculating the probability that ...
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what kind of probability distribution can be used to model numeral-noun combinations?
I'am learning German (any other language will do).
I choose randomly a countable noun - for example the noun "Hotel".
I write the noun into google together with German numeral for "two" in double ...
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Chance of something being fixed [closed]
I'm fixing a software defect that occurs 1 in n test runs. If I want to know that the probability of it being fixed is >= p for some 0 <= p < 1, how many times, m, do I need to run the test ...
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The mathematics of Schellings segregation model
For those who don't know the model. You can read this pdf. I want to find what is the probability that 2 nodes are each others neighbors when the algorithm converges (i.e. when all nodes are happy).
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Risk dice roll probabilities [closed]
Hi,
I'm wondering what probablity can tell me about dice rolling strategies in the game Risk. When a player attacks another player, they can roll up to 3 dice. The defending player can choose to ...
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Covariance and standard deviation relationship
I would like to know if an increase in the covariance between two variables would imply that the standard deviation for one of the variables has increased?
This is assuming that the standard ...
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Random projection and finite fields
Suppose we have, say, $n$ $2n$-dimensional linearly independent vectors over $\mathbb{F}_2$. We do a projection on a random $d$-dimensional subspace. We are interested in probability that images of ...
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understanding some derivation in random XORSAT problem
This question is concerned about the paper "The 3-XORSAT threshold" by O. Dubois, J.Mandler. Here is the link: http://dx.doi.org/10.1016/S1631-073X(02)02563-3
Basically one would like to know when is ...
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simulating chances of success when drawing from a bag of biased coins
I am trying to plot the pdf of flipping heads when drawing from a bag of biased coins. Since I am interested in the % of heads flipped, not the number, I simulate 500K flips and group the results into ...
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Percolation Model and Complex Probabilities
Let $d>0$ be an integer and consider the first neighbors independent bond percolation model in $\mathbb Z^d$, where each edge is open with probability $p\in[0,1]$.
I would like to know, if can we ...
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Big picture concerning Ito integral, Stratonovich integral and standard results in probability theory
I am confused and don't get the big picture concerning the connection between
Ito integral
Stratonovich integral
Standard results in probability theory concerning skewed distributions.
Example: Take ...
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How would one extend the Brier score to an infinite number of forecasts?
Is there a neat way to use something like the Brier score to score an infinite set of forecasts/outcomes?
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"Nice" Solution to repeated integral
I have a problem wherein I have defined a function $I_r(t) = \int e^{(2r-1)at} \int e^{(2r-3)at} \cdots \int e^{at} dt\cdots dt$, and $I_r(0) = 0$, for $r = 1,2,3,\ldots$.
I find that $e^{-ar^2t} I_r(...
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How can I calculate the expected ranking of a competitor from the probabilities of each competitor reaching first place?
Say I have several competitors contending over some prize. I know the probabilities that any particular one of them will win the prize. It is assumed that the competitors all want to achieve the ...
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Dynamic programming principle (DPP)
In stochastic control problem, one shall use the measurable selection theorem to prove DPP. It was discussed in discrete time case in [Bertsekas and Shreve 1978]. Is there unified framework in ...
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Good textbooks on probability and/or stochastic processes, emphasizing simulation
Any recommendations for textbooks on probability and/or stochastic processes that emphasize simulation? I'll be teaching this course in the Fall.
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"Induced" arrivals in an M/M/1 queue?
I'm a newcomer to the realm of queueing theory, so please bear with me :)
I'd like to model web server traffic with a modified M/M/1 queue.
In the simple case we have two parameters - $\lambda$ for ...
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Estimate rate of real correct/wrong from 4 answers quiz.
I recently read that one in ten students think the first man on the moon was Buzz Lightyear, a Toy story cartoon. I'm not here to discuss the data in itself, rather, this reading got me into a problem ...
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Spectrum of a generic integral matrix.
My collaborators and I are studying certain rigidity properties of hyperbolic toral automorphisms.
These are given by integral matrices A with determinant 1 and without eigenvalues on the unit circle....
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Distribution under operations
Let $X$, $Y$, $Z$, and $W$ be i.i.d. copies of a standard gaussian variable, that is in distribution $\mathcal{N}\left(0,1\right)$, then what is the distribution of $\left|\frac{XY}{Z}-W\right|$?
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Methods for choosing a result from a multiple output node Neural Network
I have a MLP with multiple nodes in its output layer which is predicting membership of classes, one class per output node. I am currently using a "winner takes all" rule for determining which output ...
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probability puzzle - selecting a person
there are n people on a round table. one of the them is the head and he plans to make another person from the rest the new head. he has a coin. he flips the coin. if he gets a head he gives the coin ...
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Formal definition of 'useful' ?
Has anyone worked out a formal, general-enough definition of what is 'useful', so that it could reflectively be used in mathematics? I am aware of the work in utility theory from economics (but ...
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literature on "stratified simulation"
I've thought of an approach to variance reduction that surely can't be new, but I haven't been able to find it published anywhere; I'd appreciate some leads.
Consider some sort of random variable $X$ ...
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Brownian Bridge under observational error
Suppose that $Z_t$ follows a simple discrete random walk $Z_t=Z_{t-1}+e_t$ , where $e_t$ are a bunch of uncorrelated normal variables with arbitrary variance sigma^2, and that there are observations ...
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Entropy of Markov processes
Consider a Markov process $X_t$ with generator $L$ and invariant distribution $\pi$, whose distribution at time $t$ is given by $\pi(t,dx)=\phi(t,x) \pi(dx)$ - in other word, $\phi(t,x)$ is the ...
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Is there a theory on two sequences of measures weakly asymptotic to each other?
Suppose $(P_n)_{n\ge 1}$ is a sequence of probability measures on a metric space $E$. Everybody knows what weak convergence of $P_n$ to a measure $P$ is.
Instead, let $(Q_n)_{n\ge 1}$ be another ...
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Random generation of subsets using conditional probabilities
Edit: Rewritten with motivation, and hopefully more clarity.
I'm building a site for a card game called dominion. In it, people build 'decks' of 10 distinct cards from a set of (currently) ...
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Can you explain a step in an expectation maximization algorithm in a Nature article?
I am currently going through the following article: http://www.nature.com/nbt/journal/v26/n8/full/nbt1406.html
In this article, how did they arrive at the values in the Estimation step (Figure 1 Step ...
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Problem with a Long Range Correlated Time Series
Consider a stochastic process $X_t$ , $t \in 1,2,3,..,N $.
$X_t$ is a Bernoulli variable and $\Pr(X_t=1) = p$ for all $t$.
The Autocovariance function $\gamma(|s-t|)= E[(X_t - p)(X_s -p)]$ is given
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Peakedness of multimodal distributions
In Probability theory, does there exist some measures of peaked-ness for multi-modal distributions. I guess kurtosis as such would not be a good measure of peaked-ness for multimodal distributions. ...
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distribution of degree of minimum polynomial for eigenvalues of random matrix with elements in finite field
This is an attempt to extend the current full fledged random matrix theory to fields of positive characteristics. So here is a possible setup for the problem: Let $A_{n,p}$ be an $n \times n$ matrix ...
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Does central limit theorem hold for general weakly dependent variables?
Say I have $X_{ij}$, $j \le i$ with the property that $X_{ij}$ are centered and identically distributed and $E(X_{ij} X_{ij'}) = o(\exp(-i)))$. Then does $\sum_j X_{ij}$ have Gaussian domain of ...
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Reference request for a "well-known identity" in a paper of Shepp and Lloyd
I ran into a "well-known identity" on page 345 of Shepp and Lloyd's On ordered cycle lengths in a random permutation:
$$\int_x^{\infty} \frac{\exp(-y)}y dy = \int_0^x \frac{1-\exp(-y)}y dy - \log x - \...
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Is the average first return time of a partitioned ergodic transformation just the number of elements in the partition?
For some reason my thinking is very fuzzy today, so I apologize for the following rather silly question below...
Let $T$ be an ergodic transformation of $(X,\Omega, \mathbb{P})$ and let $X$ be ...
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Motivation for strong law of large numbers
I always find the strong law of large numbers hard to motivate to students, especially non-mathematicians. The weak law (giving convergence in probability) is so much easier to prove; why is it worth ...
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