Questions tagged [statistics]
Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory and other branches of mathematics such as linear algebra and analysis.
11,528
questions with no upvoted or accepted answers
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Expectation of maximum of minimums of permutations
Assume $n$ random permutations $\pi_1,\pi_2,\ldots,\pi_n: \lbrace 1,2,\ldots,m \rbrace \rightarrow \lbrace 1,2,\ldots,m \rbrace$. Let $X_i = \min(\pi_1(i),\pi_2(i),\ldots,\pi_n(i))$ and $Y = \max(X_1, ...
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Expectation of truncated log-normal
Let's assume that $y=e^x$, where $x\sim N(\mu,\sigma^2)$, that is, $y$ follows a lognormal distribution.
I'm interested in finding how $\mathbb{E}\left[y|y\geq a\right]$ varies with $\mu$ and $\...
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Find a function such that follows to normal in distribution
Suppose that $X_{n}\sim \text{Binomial}(n,\theta)$, where $n=1,2,\ldots$ and $0<\theta<1$. Find a function $g$ such that $\sqrt{n}(g(\frac{1}{n}X_n)-g(\theta))\xrightarrow{D} N(0,1)$ for each ...
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An estimator for the c.d.f $F$ at a point $x_0$?
Problem: Let $X_1,X_2,\ldots,X_n$ be independent identically distributed random variables (i.i.d's) with common CDF $F$. Fix $x_0\in\mathbb{R}$ and find an unbiased estimator for $F(x_0)$. Show that ...
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The distribution of the ith order statistic for discrete random variables
Assume $(X_i)_{i=1,...,n}$ are a sequence of real iid random variables with continuous density $p_x$.
We know that $$Y:=\sum_{i=1}^n 1\{X_i\leq u\}\sim Bin(n,F_x(u)),$$ since $1\{X_i\leq u\}\sim Ber(...
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Decrease of entropy when iterating a random discrete function
Let $m$ be a positive integer. Let $S$ be the set of non-negative integers $x$ less than $m$, with $|S|=m$. Let $X_0$ be the discrete uniform distribution over $S$, with $P(x)=\begin{cases}
1/m & \...
- 1,768
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8k
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What is the Fisher information for the parameter $\theta$ in a uniform distribution with likelihood $f(X,\theta)=\frac1\theta 1\{0\le x\le\theta\}$?
If X is U[$0$,$\theta$], then the likelihood is given by $f(X,\theta) = \dfrac{1}{\theta}\mathbb{1}\{0\leq x \leq \theta\}$. The definition of Fisher information is $I(\theta) = \mathbb{E} \left[ \...
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Smallest eigenvalue of matrix with random elements (non-central Wishart)
Suppose that $X \in \mathbb R^{d \times n}$ is a random matrix with independent entries, each of which follows the standard normal law $\mathcal N(0, 1)$, and that $M \in \mathbb R^{d \times n}$ is a ...
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The statistical average of a continuous value: $\overline{O} = \int O(x) \rho(x) dx$, but coordinate invariant
I am trying to solve a Lagrange multiplier problem for the following equation
$$
L= - \int_{-\infty}^\infty \rho(x) \ln \frac{\rho(x)}{q(x)} dx + \alpha \left( 1- \int_{-\infty}^\infty \rho(x) dx \...
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How to lower bound $\tau$ based on the expression of $H$?
Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ normalized symmetric Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots \...
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Is the Fisher-Information even continuous in a regular statistical model?
Definition (Regular Model [1, p. 203]).
A standard statistical model $\big( X, \mathcal F, (\mathbb P_{\vartheta})_{\vartheta \in \Theta}\big)$, where $\Theta \subset \mathbb R$ is an open interval, $...
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245
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Why doesn't the Borel-Kolmogorov paradox cause problems in practice?
The Borel-Kolmogorov paradox shows that the usual formula for conditional density $f_{X|Y}(x|y) = f_{X,Y}(x, y)/f_Y(y)$ can lead to inconsistent results depending on the coordinate system that is used ...
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143
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Correct measure in concentration inequalities or hypothesis testing
In most discussions of concentration inequalities or calculations of rejection region in hypothesis testing, the measure used is left vague. For example, for independent random variables $X_1, \ldots, ...
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Is there a pattern to the coefficients in the piecewise equations of the Irwin–Hall distributions?
Intro and Problem Statement
The Irwin–Hall distribution is a probability distribution of the sum of $n$ independent, uniformly-distributed, continuous random variables in the interval $[0, 1]$. The ...
- 1,861
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365
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Estimate nearly-singular Gaussian covariance matrix
Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need sample covariance $\Sigma_m$ to be $\epsilon$-close to true covariance $\Sigma$:
$$E\|\Sigma_m-\Sigma\| \le \epsilon \|\...
- 5,489
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Creating a Skyrim Alchemy metric: combining the binomial theorem and birthday paradox
I am trying to create a metric to sort alchemy ingredients by in Skyrim, I wanted to sort the ingredients by "probability of yielding a potion with the number of ingredients I have". The ...
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390
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What is the expected value of the inverse of a Wishart matrix plus a scaled identity matrix?
I'm trying to find the following:
$$\mathbb{E}\left[\left(\mathbf{W}+\lambda\mathbf{I}\right)^{-1}\right]$$
where $\mathbf{W}\sim\mathcal{W}_{p}(n,\mathbf{\Sigma})$ is Wishart-distributed, $\lambda\ge ...
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Book recommendation for convergence (Almost Surely, probability, distribution) concepts in statistics.
I want to study the convergence concepts in statistics. I am mainly interested in topics like
Convergence in Probability
Convergence in Distribution
Almost Surely convergence
I can't seem to find ...
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Given a derived random variable $Z=f(X_1,\dots, X_n)$ on $X_j$, what can we say about $X$?
Let $Z$ be a derived random variable given by some function $f$, i.e.
$Z=f(X_1,\dots, X_n)$, where the $X_i\sim X$ are continuous/non-atomic and independently and identically distributed. What can we ...
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186
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About solving $f(z) = \frac{1}{4} (f(-1-z) +2 f(-1-2z)+ 2f(-1+2z) +f(-1+z))$
Final update on 11/29/2019: I have worked on this a bit more, and wrote an article summarizing all the main findings. You can read it here.
I am looking for a solution where $f(z)\geq 0$ and $\int_{-\...
- 3,645
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2
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731
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Distribution of sample variance of Cauchy distributed variables
Assume $X_i,i\in\left\{1,...,n\right\}$ are i.i.d. standard Cauchy distributed random variables.
I know that $\bar{X}_n:=\frac{1}{n}\sum_{i=1}^n X_i$ is standard Cauchy distributed.
I would like to ...
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Show that the Normal distribution is a member of the exponential family
I want to show that the Normal distribution is a member of the exponential family.
I have been working under the assumption that a distribution is a member of the exponential family if its pdf/pmf ...
- 257
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Proof of a technical fact in the book of Schapire and Freund on boosting
I am currently looking at Exercise 10.3, Chapter 10 of the book on Boosting by Schapire and Freund. More precisely, in the middle of the exercise they propose to use, without proof, the technical fact ...
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8k
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When to use cumulative moving average vs a simple moving average?
After reviewing the Wikipedia page on moving averages, the difference between the simple moving average and cumulative moving average are clear:
1) Simple moving average only considers the last n ...
user541285
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Variance of the Euclidean norm of a vector of Gaussians
Given a vector of correlated Gaussian random variables $\vec{X}=\left(X_1, ..., X_k\right)$ that are normally distributed $\vec{X} \sim \mathcal{N}\left(\vec{\mu}, \Sigma\right)$ with arbitrary $\...
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Area of a triangle where the three vertices are randomly chosen on a circle; also $3D$ version.
My teacher gave us an interesting problem today. Consider a circle of radius $1$, choose three points on that circle at random and make a triangle connecting the three. On average what will the area ...
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664
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Efficient way of integrating a 2-dimensional Gaussian over a convex polygonal domain
I am looking for a reference for the calculation of integrals of bivariate normal distributions over (convex) polygons.
$$P\{X\in\mathcal{A}\} =
\int_{\Omega} \frac{1}{\sqrt{(2\pi)^2|\Sigma|}}e^{-\...
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400
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Proof of multivariate change of variable technique in statistics
I am having hard time conceptually grasping how the bivariate change of variable technique works in statistics. The technique can be summarized as follows: Given random variables $X$ and $Y$, and ...
- 1,064
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583
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Radon-Nikodym on a Process wrt to filtration
Given a probability space $(\Omega,\mathcal{F},P)$. Let $(X_t)_{t\geq0}$ be a stochastic process defined on it with cadlag paths, lets say on $(\mathcal{X},\mathcal{B}(X))$.
Let be $\mathcal{F}_{t}$ ...
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6
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References for information theoretic statistical tools
Statistical concepts like spaces of probability distributions, "metrics" like Fisher information or relative entropy, and convergence with respect to these quantities are needed in my ...
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