All Questions
Tagged with pr.probability st.statistics
1,134 questions
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Distribution for probability of an incorrect inference based on a comparison of only two samples?
I'm trying to demonstrate the problems of how taking a sample and assuming it reflects the population accurately can be problematic.
Imagine say an urn with some large number of balls, black and ...
- 11
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2
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362
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A machine learning application question
I am familiar with basic probabilities, random processes but not so much of machine learning methods. This is the problem I am trying to solve.
I want to predict the nature of user activity on a ...
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442
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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 ...
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3
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Probability that one RV will exceed many others
Assume the $1 \times N$ vector
$\mathbf X = [X_1, X_2, \ldots , X_N]$
contains i.i.d. normal samples such that $\mathbf X$ has a multivariate normal distribution. Now assume another random variable $...
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Probability distribution for two-state system that depends on residence time
I am a statistical physicist, and I've come across a problem that I don't know how to solve. I believe my issue lies with how to formulate it mathematically. I'd be very grateful for any assistance, ...
- 188k
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Transition time in finite voter model
I believe the following problem is related to something called the "voter model" in statistics. This is not my area of expertise so please forgive me if the answers turn out to be well known.
...
CommunityBot
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Expectation of the trace of an inverse of a random matrix
Given a $N \times M$ matrix $X$ comprised of standard normal entries ($M > N$), I'm interested in approximating $E[trace((XX^T\frac{\gamma}{M} + I)^{-1}]$ in terms of $N, M$ and $\gamma$. ...
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A simplified MCMC / MH algorithm. Are there known convergence results?
Hi, I hope this isn't too basic. We were working on a simulation using a Monte Carlo Within Metropolis algorithm and noticed that the whole thing could be expressed in the form below and simplified ...
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Examples of Slowly Mixing Chains in Statistics
This should probably be community wiki, but I don't know how to set that myself.
I'm looking for examples or Markov chains that are used in statistics or statistical physics, and which are known to ...
CommunityBot
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160
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Two Different Representations of Multivariate Bernstein Polynomials
In the literature the multivariate Bernstein polynomial of a function $f:[0,1]^m\rightarrow\mathbb{R}$ is often defined as the following:
$$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k}\in \{0,\dots,n\}^m}...
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Is the Binomial Expectation of Convex Function Convex in p?
Suppose $X$ has a binomial distribution with success probability $p$ and $n$ trials and let $h(\cdot)$ be a positive convex real-valued function.
Is the function $g(p)=\mathbb{E}[h(X)\ |\ p]$ convex ...
CommunityBot
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245
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Random walk conditioned on sum and last step
Can anyone help with the following (slightly weird) random walk question? I have a random walk starting at $X_0 = 1 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$. ...
CommunityBot
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789
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Uniform law of large numbers for martingale difference
Let $\xi_{tn}(\theta),t=1,\dots,n$ be a real-valued martingale difference array indexed by a parameter $\theta \in \Theta \subset R$, where the set $\Theta$ is compact. Now, for all fixed $\theta \in \...
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Derivative of a random process
Consider $w(t)$ as Guassian random process, with $w(t)$ being $\mathcal{N}(\mu,\sigma)$ and i.i.d for all t.
I consider applying a (stochastic)derivative operation to the random process. What is the ...
- 3,069
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calculating how much to oversell given an acceptable risk (statistics)
I have a shared resource with a finite capacity (let's say 100), and I have usage data (2 hours average of samples taken each 20 seconds). I accept a risk of 10% per year to reach the capacity.
...
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Do there exist (almost surely) $C^{\infty}$-smooth Gaussian random fields?
Let $d \ge 1$. Do there exist Gaussian random fields on $\mathbb R^d$ which are (almost surely) $C^{\infty}$-smooth, but which are not analytic?
If so, what are necessary and sufficient conditions ...
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298
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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)$$
for all pairs $(X_i, X_j)$, does ...
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112
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Markov renewal process with failure?
I hope this question is not too elementary for this site, and that it contains a sufficient degree of detail.
I have a problem where I want to model sequences of variable length $\boldsymbol{e}_i = (...
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1
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256
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Taking the partial derivative of the t-CDF with respect to the degrees of freedom
I am trying to find the maximum likelihood estimate of the parameters for the t-copula. Ideally I'd want to use a gradient-based method for optimization. However, I am having some difficulty in ...
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262
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Given that a conditional measure is Gaussian, how bad can the original measure be?
Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...
- 8,512
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200
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Moments of random matrices - when are they finite
I need to evaluate the moment
$$\mathbb{E} (AX)^n,$$ where A is an NxN Hermitian square matrix, and X is
$$X=ZZ^{\ast},$$ where
$Z=\mu+Y$, where $\mu$ is mean of $Z$ and $Y$ is a zero-mean complex ...
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108
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"Soft" Voronoi cells or statistical criterias
It is probably some basic statistics question, but...
Informally 1: How to choose "criteria", such that it will guarantee that error decision probability is less than "epsilon", and maximize ...
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3
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673
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convex combination of two covariance estimates
I am interested in covaraince matrix estimation. In brief: I have two estimates of the covariance matrix, and now I want to form a bona fide convex combination of the two.
Background: I have studied ...
CommunityBot
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577
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Prove an inequality related to moments
I am reading a paper and stuck with an inequality used in that paper.
$\varepsilon^n=(\varepsilon_1^n, \varepsilon_2^n,\ldots,\varepsilon_n^n)^T$ is a vector of i.i.d. random variables with mean 0 ...
- 114k
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1
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Applications of the Giry monad in probability and statistics
In another thread, I asked about the $M$ endofunctor on the category $\operatorname{Meas}$ of measurable spaces, which sends a space $X$ to its space of measures $M(X)$.
Will Sawin described the ...
CommunityBot
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280
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Deciding whether or not an exponentially distributed random variable exists in a set via the use of a "black box" function
I have some set of known size but with unknown elements, $(x_1, ..., x_N) \in X$, where the elements of $X$ are exponentially distributed random variables with unknown rate parameters, $(\lambda_1, ......
CommunityBot
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What's the maximum entropy probability distribution given bounds [a,b] and mean?
What is the continuous probability distribution that maximizes entropy, given only the bounds of the random variable [a,b] and the mean mu of the probability distribution?
For example:
if a=0, b=1, ...
- 148k
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153
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Sampling without replacement: probability for total successes from successes in sample?
Consider drawing $n$ balls from an urn containing $N$ balls, of which $m$ are red. If i know $N$, $m$ and $n$ i can use the hypergeometric distribution to calculate the probability that my sample ...
4
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1
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213
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Practical way to check for geometric convergence
Target distribution is multimodal, 24 dimensions, continuous state space. For MCMC integration (MH sampler) I use a manually tuned proposal distribution.
When I measure the convergence rate ...
- 1,302
2
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1
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421
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Extending Wald's equation to two classes of i.d. random variables?
I try to adopt Wald's equation to a slightly more complex problem. In fact, after a full day, I found some solution now, but it has a confusing argument in the middle. Perhaps somebody can help me at ...
CommunityBot
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213
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Find a minimum entropy code for a simple gibbs random field.
Just to make precise what I am talking about, I will include the definition of a minimum entropy code. I will then define the precise markov random field I am asking about.
In the rest of this ...
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368
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Product of probability densities of the form x^{-t} exp (-ax)
I have two probability distributions $p(x) = N_1 x^{-\tau} \exp(-\frac{x}{x_0})$ and $p(y) = N_2 y^{-\kappa} \exp(-\frac{y}{y_0})$. $N_1$ and $N_2$ are just normalization constants and $x>0$, $y>...
- 54.2k
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Kalman Filter...Denoising measurement data to track objects
Hi Everyone,
I am about to implement a Kalman Filter in a software.
I found this very helpful article here:
http://bilgin.esme.org/BitsBytes/KalmanFilterforDummies.aspx
The example helps a lot, ...
- 1,302
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265
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Expectation of a multivariate Gaussian over a plane
For a vector $X$ which follows a multinomial Gaussian distribution $N(\vec{0},\Sigma)$, a given vector $b$, and a known scalar value $c$, I would like to calculate the expectation :
$E[X|X^Tb = c]$
...
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2
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2k
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Total variation distance between a Poisson and a distribution with known mean/variance
Suppose that $\mu$ is the law of a Poisson distribution of mean 1, and that $\nu$ is the law of an unknown distribution on the non-negative integers, though I do know that its mean and variance are ...
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107
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Can one combine (join) probabilities from 2 aspects of a related process?
Consider 2 related aspects of a process for prices in a financial market:
time &
return.
Time
Say I've identified a distribution that reasonably models the occurrence of the lengths of price ...
CommunityBot
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1
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101
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multimodal circular model
Hi, can someone provide me with a list of probability models that is akin to Von Mises but consists multiple (potentially infinite) modes that takes into account attractors in the entire 2-D spatial ...
CommunityBot
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4
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Recent impressive combinatorial developments in probability theory
In the preface to the second edition of Daniel Stroock's book "Probability Theory: An Analytic View", there is this striking claim (on p. xv)
... I suspect that, for at least a decade, the most ...
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447
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MCMC with progressive demollification of delta distributions
Edit: I simplified the example to a canonical case for clarity.
Given an integral $\int_{\Omega}{g(\mathbf{x})}$ with a well-posed integrand $g(\mathbf{x})$ defined on some multidimensional space $\...
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prewhitening (whitening transform) in terms of expected-value-wr-sigma-algebra
I'm trying to understand the mathematics of prewhitening a little better. (See http://en.wikipedia.org/wiki/Whitening_transformation, e.g.)
Taking the conditional expectation of an RV with respect to ...
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Limiting distribution of the cardinal of a Markovian set
Let $S_1=\lbrace u_1 \rbrace$ where $u_1$ is a random uniform drawing on $[0,1]$. To build $S_{n+1}$ draw $u_{n+1}$ uniformly on $[0,1]$ (independently from previous draws) and draw $v_{n+1}$ ...
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154
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Positive estimator
Suppose that one knows how to generate (independent) random samples $X_1, X_2, \ldots$ distributed as the random varable $X$ with $\mathbb{E}[X]=\mu \in \mathbb{R}$. It is then easy to construct an ...
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Are Gaussian Processes more important than other stochastic processes?
I am doing a course at university and it deals with Gaussian Processes mainly. We use them for fitting data and prediction, machine learning, regression, classification. Is there any particular reason ...
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Conditioning over Conditional probability? (also: $\phi$-mixing sequences)
For two sub $\sigma-$fields $\mathscr{F}$ and $\mathscr{G}$ of a probability space $(\Omega , \mathscr{A} , P)$ we define $\phi$ mixing as follows:
$$
\phi(\mathscr{F},\mathscr{G}) = \sup \{ |P(G|F) - ...
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Comparing distributions with moments
Suppose I have two variables $X$ and $Y$ which have continuous p.d.f.s $f$ and $g$ on the positive real line. I know that the moments $\mathrm{E}[X^n] > \mathrm{E}[Y^n]$ for sufficiently large $n$ (...
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271
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Convergence of sample mean
I have a two-index succession of real-valued random variables $x_{t,n}$ such that $\lim_{n\to\infty} x_{t,n} = x_t$, for all $t$ and suitable limit r.v. $x_t$.
I would like to prove that $$\lim_{n\to\...
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203
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Bounds on tails with moments
A sort of continuation of Comparing distributions with moments
Suppose I have some estimates of the moments of a non-negative random variable $X$: $$\log \mathbb{E}(X^n) = n \log n + (\beta-1)n + O(\...
CommunityBot
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4
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854
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Laplace transform on the cone of positive-definite matrices
The title says most. Let $P_p$ be the cone of positive-definite $p \times p$ matrices.
One can define the Laplace transform of (the distribution of) a random matrix with values in $P_p$ by (for ...
1
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1
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281
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A uniqueness proposition involving Erf, the error function
This is a generalization of a previous MO question, "Reducing system of equations involving Erf, Error Function".
Consider the system of equations:
$$1/2 + {\rm Erf}(x) - \alpha {\rm Erf}(\frac{x+y}{...
CommunityBot
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2
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Reducing system of equations involving Erf, Error Function
I have a system of equations:
$$1/2 + {\rm Erf}(x) - {\rm Erf}(\frac{x+y}{2})=0$$
$$-1/2 + {\rm Erf}(y) - {\rm Erf}(\frac{x+y}{2})=0,$$
Where $x \le y$ and ${\rm Erf}$ is the Error Function.
By ...
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