All Questions
Tagged with pr.probability st.statistics
1,134 questions
4
votes
2
answers
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views
Conditional Probabilities - The Mad Kings' Draft
The Problem of the Mad King's Draft:
Suppose there is country which is ruled by a king who can be either 'mad' or 'normal.' The king rules a a large country with a continuum of citizens who have ...
3
votes
0
answers
125
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Is a parametric family which is universally consistent for multiple quantiles impossible?
Suppose I am dead-set on using Bayesian inference on independent and identically distributed data, but I'm lazy and insist on using a parametric likelihood function come what may. I'd be reassured to ...
- 2,791
1
vote
1
answer
394
views
Conditional probability and independence
Suppose that we have vectors of events $\{H_1,...,H_n\}$ and $\{D_1,...,D_m\}$. Consider the following two sets of conditions:
Condition set 1
(1) $P(H_i H_j)=0$ for any $i\neq j$ and $\sum_iP(H_i)=...
- 1,902
1
vote
2
answers
772
views
Gibbs sampling step size
I have some data generated using MCMC methods and in particular Gibbs sampling. I computed the autocorrelation but I'm unsure how to determine how many samples to skip.
I'd like to determine that ...
- 2,791
3
votes
1
answer
4k
views
Derivatives of conditional expectations
Let $X$, $Y$ and $Z$ be independent, real-valued random variables, probably with continuous density functions. Define $A = X + Y$ and $B = X + Z$. Consider the regular conditional expectation $E_Y(a,...
- 46
1
vote
0
answers
1k
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Distribution of uniform-normed random vector
What is the pdf of $\vec{Y} = \frac{\vec{X} }{\lVert \vec{X} \rVert_\infty}$ with $\vec{X}$ a random vector following a multivariate standard normal distribution (zero-mean $\vec{\mu} = 0$ and ...
- 21
3
votes
0
answers
143
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finding rank-3 tensors compatible with a rank-2 tensor projection
I am interested in the following problem: Consider a rank-3 symmetric tensor $\boldsymbol{\sigma}$ with $\sigma_{ijk}$ where $\sigma_{ijk}$ can be 0 or 1, and the symmetry is with respect to any ...
- 41
4
votes
1
answer
151
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Mean occurrences of letters in complete strings given by a Bernoulli scheme
Suppose one has an alphabet of $K$ letters, from which we draw sequentially letters; assume that the $n$-th letter occurs with a fixed probability $p_n$ independently of the others and of the previous ...
- 96.4k
0
votes
1
answer
207
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Copulas and marginals thereof
Hello everyone,
I recently became aware of the existence of the copula concept.
So, I have been reading a few things about copulas lately, but
I cannot seem to find information on the following ...
- 1,902
1
vote
0
answers
336
views
Normalized correlation with a constant vector
I am confused how to interpret the result of preforming a normalized correlation with a constant vector. Since you have to divide by the standard devation of both vectors (reference: http://en....
- 171
3
votes
2
answers
334
views
Scale random variables in a way they have equal probabilities of being minimal
I have several positive random variables $x_i,\ i=1,...,N$ taken from different unknown distributions (these distributions can be closely approximated by log-normal if needed). I can sample these ...
- 32.4k
2
votes
0
answers
1k
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Random variables: multivariate second-order Taylor approximation (delta method)
Let $g:\mathbb{R}^2\rightarrow \mathbb{R}$ be a smooth, but not necessarily bounded function and $X$ and $Y$ two random variables that are not independent. (assuming they yield sufficiently many ...
- 21
7
votes
3
answers
995
views
Kolmogorov probability axioms without non-negativity condition
What is a minimal consistent modification of probability axioms to include negative values?
Is it enough to use a minimal modification of axioms obtained by
formal exclusion of non-negativity ...
- 19
2
votes
0
answers
422
views
Generalizations of Gram-Charlier and Edgeworth series?
I am looking for references pertaining to, and/or help in deriving, generalizations of the Gram-Charlier and Edgeworth series for non-Gaussian reference probability distributions.
I would like to ...
- 1,890
1
vote
1
answer
282
views
Is an unbiased estimator with arbitrarily small variance necessarily consistent?
Given an unbiased estimator $\hat \theta_n$ of a parameter $\theta$, if the estimator has small variance (approaching $0$ as $n\to\infty$), it seems reasonable to expect that the estimator is ...
0
votes
1
answer
577
views
Expectation of little o in probablity [closed]
If I have $Z=o_p(1)$ where $o_p$ is the little-o in probability. I'm interested in find some properties about $E(Z)$.
My first idea was
$E(Z)=E(Z (1_{Z>\varepsilon} + 1_{Z\leq\varepsilon}) ) \...
CommunityBot
- 1
0
votes
1
answer
666
views
A Cauchy–Schwarz Type Inequality Involving Scaled Distributions
I have stumbled upon a rather intriguing inequality involving the product of the scaled distribution and the scaled density of a random variable. The inequality has a very attractive form, and it ...
- 197
1
vote
1
answer
3k
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Convergence of Eigenvalues
Suppose we have a matrix $A_n = \frac{1}{n}\sum_{i=1}^nX_i X_i^T$, where $X_i$ is a $p$-dimensional random-vector. We also know that $E(XX^T) = \Sigma_{p \times p}$. Let us denote the $j$-th largest ...
CommunityBot
- 1
8
votes
1
answer
2k
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Eigenvalue distributions of finite dimensional Wishart matrices
I am trying to obtain the eigenvalue distribution of a finite dimensional Wishart matrix. Let $A_{n\times n}\sim\mathbb{W}(\Sigma_{n\times n},m)$ where $\mathbb{W}(\Sigma_{n\times n},m)$ denotes the ...
CommunityBot
- 1
1
vote
1
answer
302
views
Log concavity of noncentral chi-square
I am trying to check if the noncentral chi-square distribution is log-concave in its noncentrality parameter. Specifically, given
$p(y ; \lambda, \sigma^2) = \frac{1}{2\sigma^2}\exp\left(-\frac{y+\...
CommunityBot
- 1
1
vote
0
answers
236
views
density for Gaussian gram matrices
Let $Z \sim \mathcal{N}(0,\Sigma \otimes I)$ (so the columns of $Z$ are distributed $\mathcal{N}(0, \Sigma)$) and $A = Z'Z.$ Is there a name for the distribution on $A$? Is the density known?
- 922
3
votes
0
answers
206
views
representing vine copulas
Vine copulas is a way to represent multidimensional distributions (n-densitys)
as a product of the n 1-marginal densities and a product of (n choose 2) bivariate copulas, where som of them are ...
- 2,600
2
votes
0
answers
979
views
How to calculate/approximate expectation of function of a binomial random variable?
Hi,
I am stuck at following problem in my research.
Suppose that $M=m$ is a random variable with binomial distribution with parameters $n,p$. The constants $r$ and $\gamma$ are greater than zero. $\...
- 21
0
votes
1
answer
285
views
Is there a monotone coupling of Dirichlet random variables?
Let $X=(X_1,X_2,X_3)\sim \text{Dirichlet}(a_1,a_2,a_3)$ and $Y=(Y_1,Y_2,Y_3)\sim \text{Dirichlet}(a_1+b_1,a_2+b_2,a_3)$, where all $a_i$ and $b_i$ are positive. Is there a natural coupling between $X$ ...
- 6,711
4
votes
3
answers
467
views
Law of large numbers for stochastically chosen samples
Let $X_t$ be a sequence of i.i.d. random variables with mean $\mu$. Then the law of large numbers states that
$$\lim_{T \to \infty} \frac1T \sum_{t=1}^T X_t = \mu \quad a.s.$$
Now suppose that (in a ...
- 103
1
vote
1
answer
5k
views
Distribution of the standard deviation of normal variates
What is the distribution of the standard deviation of $n$ normal variates? That is, if $X_1,...,X_n$ are i.i.d. normal random variables with mean $\mu$ and s.d. $\sigma$ and $M=\sum X_i/n$, then what ...
- 96.4k
6
votes
1
answer
4k
views
Conditioning on one term of a sum of random variables
Let $\theta$ be normally distributed with mean $\bar \theta$ and variance $s^2$. Let $Z$ be normally distributed with mean $0$ and variance $\sigma^2$, and chosen independently of $\theta$. Define $...
- 8,512
1
vote
1
answer
742
views
proofs of stochastic boundedness
I'm looking at some statistical literature and trying to compare the results given there in probabilistic big-Oh notation with statements I'm more familiar with.
In particular, I'm trying to ...
- 6,711
0
votes
2
answers
327
views
Copulas and time series
Please, can anybody give a reference(s) to some good recent review papers about copulas and time series?
- 1,334
1
vote
1
answer
502
views
nonnegative series expansion of nonnegative functions
The title says it all! When using orthogonal series expansions like the Gram-Charlier expansion to approximate probability density function, a big problem (making this approach less usefull and less ...
CommunityBot
- 1
3
votes
0
answers
171
views
Iterated Kumaraswamy distributions
The Kumaraswamy distribution has cdf $F(x;a,b) = 1-(1-x^a)^b$.
Does anyone know any formulas or properties relating to iterations of this on itself, meaning
$$ F_i(x;a,b) = 1-(1-F_{i-1}^a)^b$$
If ...
- 233
0
votes
4
answers
386
views
Recovering a function from a set of approximations
We assume that we have a finite set of agents with approximate knowledge about a certain function, and from this collection of approximations we want to recover the actual value of the function.
More ...
CommunityBot
- 1
2
votes
3
answers
403
views
On a randomized version of compressive sensing
The compressive sensing theory of Candes and Tao (See http://en.wikipedia.org/wiki/Compressed_sensing) relies highly on the fact that the underlying data (such as a signal or an image) is sparse or ...
- 101
24
votes
2
answers
1k
views
Drawing natural numbers without replacement.
Suppose we start with an initial probability distribution on $\mathbb{N}$ that gives positive probability to each $n$. Let's call this random variable $X_1$ so we have $P(X_1=n)=p_{1,n}>0$ for all $...
- 5,721
13
votes
7
answers
1k
views
Probabilistic (and other mathematical) methods of physics without the physics?
Many of the methods of physics are vastly more general than their use in that discipline. For example, information theory overlaps with a lot of statistical mechanics, and the latter actually ...
- 513
8
votes
0
answers
4k
views
Taylor approximation of a function of a random variable
Suppose we have a random variable $X$ and a smooth function $g$. We want to calculate the expectation value $\mathbb{E}(g(X))$. To be able to write down at least an approximate solution, we perform a ...
- 91
8
votes
2
answers
1k
views
Order statistics (e.g., minimum) of infinite collection of chi-square variates?
Hi everyone,
This is my first time here, so please let me know if I can clarify my question in any way (incl. formatting, tags, etc.). (And hopefully I can edit later!) I tried to find references, ...
CommunityBot
- 1
6
votes
1
answer
1k
views
Probability distributions: The maximum of a pair of iid draws, where the minimum is an order statistic of other minimums?
General question: What is the distribution for the maximum of 2 independent draws from cdf F(x), when we know that the minimum of those same two draws is the kth order statistic of the minimum of n ...
CommunityBot
- 1
1
vote
1
answer
453
views
An infinite Gaussian mixture with mixing parameters being also Gaussian
A finite Gaussian mixture with $k$ components has a probability distribution function $p(y|\mu_1,...,\mu_k, \sigma_1, ..., \sigma_k, \pi_1, ..., \pi_k)=\sum_{j=1}^{k} \pi_j\mathcal{N}(\mu_j, \sigma_j^...
CommunityBot
- 1
3
votes
1
answer
731
views
name for $\underset{x}{\operatorname{argmin}} \displaystyle\sum\limits_{i=1}^n |x_i - x|$
Given a real-valued data set $ x_1, \dots, x_n $, what do you call the quantity
$$\underset{x}{\operatorname{argmin}} \displaystyle\sum\limits_{i=1}^n |x_i - x|$$
This seems like a pretty basic ...
- 3,475
4
votes
2
answers
653
views
Reading Material on Couplings
Does anybody have suggestions on what to read to learn more about couplings pertaining to statistics?
I'm working on a research project on Poisson approximations and am looking to perform a coupling ...
- 25.7k
9
votes
2
answers
674
views
Small crown probabilities (and infinite dimensional margin assumption)
My question is:
How do I find sharp upper bounds on $P(|q|\leq \epsilon)$ uniformly over a set of gaussian polynomes $q$ of degree two.
Notations and definitions (to make the question rigorous)
Let ...
- 96
0
votes
2
answers
257
views
Efficient computation of $E\left[\frac{1}{1+\sum_iX_i}\right]$ where $X_i$ is RV with Bernoulli distribution with different probabilities
Suppose we have the random variables $X_1, \ldots, X_n$ that have Bernoulli distributions with the (possibly different) probabilities $p_1, \ldots, p_n$. For example, $X_1$ = 1 with probability $p_1$ ...
- 47.4k
1
vote
1
answer
221
views
Estimating the Distribution of a Very Large Population of Known Size and Unknown Variance
I would like to estimate the distribution of a very large population of known size but unknown mean and variance. I cannot assume anything about the underlying distribution. The values of observations ...
- 196
15
votes
3
answers
2k
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entropy and flatness of densities
I was reading C.R Rao's Linear Statistical inference. Rao presents the entropy of a continuous distribution (expectation of -log density) as a measure of closeness to the uniform distribution, and ...
- 196
1
vote
0
answers
554
views
How to obtain tail bounds for a linear combination of dependent and bounded random variables?
Hi everyone,
Note: This question is a general case and edited version of my previous question ``How to obtain tail bounds for a sum of dependent and bounded random variables?''.
I am looking for ...
- 197
0
votes
1
answer
275
views
Conditional distribution of the modulus of the output of AWGN channel given the modulus of the input
Hi everyone,
I will be too happy if anybody help me find a solution for the following problem.
In fact, I have a big problem that I could not solve it for weeks.
Assume that we have we have two ...
- 560
3
votes
2
answers
255
views
Correcting bias in samples selected by a prediction
Here is the scenario:
I'm trying to find as many golden tickets as I can, so that I can sell them to kids that want to go on a tour of Wonka's chocolate factory.
Fortunately, I have a machine that ...
- 28k
5
votes
1
answer
1k
views
Probability inequalities
Hi everyone,
I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts.
My problem is to find an ...
- 197
3
votes
4
answers
2k
views
statistical approach to multinomial distribution
Suppose a dice with $q$ faces is rolled $N$ times, where $N$ is very big.
We define a multinomial variable $X=(X_1,\ldots,X_q)$ which counts how many times any face is occurred ($X_i$ is the number ...
- 5,721