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.
37,751 questions
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Finding an exact closed (non-recursive) form formula for the probabilities in a game of Risk
Yesterday I enjoyed some rounds of RISK: Global Domination with a friend from university. It is a long-running in-joke that “True Random” is the cause of winning and losing certain battles.
Our ...
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Double negatives in hypothesis test conclusions
I am grading hypothesis tests for an introductory statistics class and students occasionally give the following conclusion after rejecting the null hypothesis:
Since $H_0$ is rejected, there is not ...
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Covariance of Unbiased Sample Variance Estimators with Overlapping Samples
Unbiased Variance Estimator
Let $x_1 , \ldots, x_N$ be iid sampled from X.
Let Y(N) denote the N-mean estimator given by
$$ Y(N) = \frac{1}{N} \sum_{i=1}^N x_i $$
Let v(N) denote the unbiased N-...
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Predicting $Y$ from a correlated variable $X$
I was reading something. The context was we could measure the variables $X$ and $Y$ on individuals. And it appeared that $X$ and $Y$ were correlated with correlation: $\rho=0.3$.
The writer then ...
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Mixture Model Expectation
We know that if an i.i.d. sample is drawn from $p_{\theta}=\text{Ber}(\theta)$, $\theta\in (0,1)$ then
$$\mathbb{E}_{p_{\theta}}[\bar{X}] = \theta,$$
where $\bar{X}$ denotes the sample mean.
Now, ...
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Dominated statistical models
This is a simple question from measure theory.
Fix a measurable space $(E,\mathcal{E})$ and a family $(P_i)_{i\in I}$ of probability measures on $(E,\mathcal E)$ ($I$ is any non-empty set). Let $n\geq ...
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calculating reliability statistics- combined incidents per thousand units. [closed]
I would like to combine warranty information across different products. for a combined warranty metric (incidents per thousand units). Is this a case of multiplying together?
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Verifying One-vs-All Precision and Recall calculations from a multi-class confusion matrix
I am studying multi-class classification metrics and want to confirm the correct way to compute them from a confusion matrix.
A weather classifier labels days as Sunny, Rainy, Cloudy. The test results ...
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“Central limit theorem” for symmetric random variables with no finite mean
Let $(X_n)_{n\in\mathbb N}$ be a sequence of independent random variables with the same distribution. The common distribution $\mu$ is such that it is symmetric, that is, $\mu((-\infty,x])=\mu([-x,\...
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How much less is the arithmetic mean than the max given the average deviation?
Given a finite (multi)set of elements $\{x_1, \ldots, x_n\}$ the arithmetic mean $\mathsf{AM}$ is less than or equal to the maximum element call it $\max$. In otherwords, $\mathsf{AM} \leq \max$. But ...
- 3,630
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Does online mirror descent between dually flat space converge to the global optimum
Say $E = \{p_\theta : p_\theta(x) = \exp(x^\top \theta - A(\theta)), \theta \in \Theta, M \theta = b\}$ is an exponential family affinely constrained in its natural parameter, where $\Theta$ is a ...
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An MLE Asymptotic Normality problem with i.n.i.d. data
Suppose $n \in \mathbb{N}$. Suppose $s_0 > 1$ and $\xi_j \sim N (0, j^{- s_0}
+ n^{- 1})$, $j = 1, 2, 3, \ldots, n$. Let $\hat{s}_n$ be the maximum
likelihood estimator of $s_0$. Is $\hat{s}_n$ ...
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The quantile transformation in normal distribution
I'm reading the textbook, Probability and Statistical Inference by Hogg and others.
When explaining the Q-Q plot, it is said that
when $q_p$ is a quantile of the normal distribution N($\mu, \sigma$), ...
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Can mutual information be defined between a random variable and a conditional distribution
Quantities like mutual Information $I$, entropy $H$,etc. are typically defined as taking random variables as input. However, they are actually just functions on probability distributions - e.g. the ...
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Calculating how many samples are needed to get the desired confidence interval
Below is a problem I made up and tried to solve. I am hoping somebody can help me finish it.
Problem:
A magical device generates a normally distributed random number with standard deviation of $1$ and
...
- 4,622
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Log-linear model for 2 way contingency table
I have some confusion on part c of the problem.
Our null hypothesis is $$H_0:\pi_{1j}=\pi_{2j}=\pi_{3j}=\pi_{4j}\\\forall j$$
Should our log-linear model be $$logu=u+uT+uR$$ or $$logu=u+uR$$
where uR ...
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Using MLE to estimate the mean of a normally distributed random variable
Problem:
A certain set of values if known to be normally distributed with $\sigma^2 = 1$. However, its mean is not
known. The following three sample values are taken: $0, 1, 10$. We want the best ...
- 4,622
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Where can I find more information about 'specific information'
I am looking to borrow a concept from information theory. Namely, the mutual information between two random variables $X, Y$ when one of the random variables is fixed, e.g. $Y = y$:
$$
I(X ; Y = y)
$$...
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Can probability theory be made fully computable?
I’ve been reading about objective Bayesian theories lately and came upon the concept of universal priors and specifically, the Solomonoff prior. This seemed to answer my initial query about whether a ...
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A statistics problem dealing with normal random variables - My answer is close to the answer in the book - is it close enough?
Below is a problem I did. I feel my answer is right. However, it is different than the book's answer. Is the difference simple round-off error?
Problem:
The length of time required for the periodic ...
- 4,622
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the limit of variance and expectation
Let $\varepsilon_1, \dots, \varepsilon_n$ be independent random variables with $E(\varepsilon_i) = 0$.
Let $f: [0,1] \to \mathbb{R}$ be a Lipschitz function with constant $K > 0$, i.e.,
$$|f(x) - f(...
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Average distance between nearest neighbours of 𝑛 randomly distributed points on a rectangle [closed]
Assume $a$ and $b$ are sides of rectangle, I guess we need to take two random vectors, $X$ and $Y$, then $E(|X-Y|)$ will be overall expectation, how to constrain to nearest neighbours
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Pearson's Correlation Coefficient with respect to $z$-scores.
In my AP Statistics class, the coverage of Pearson's Correlation Coefficient was pretty limited. It boiled down to "it's a measure of correlation such that $\hat{z_{y}}=rz_{x}$", and he only ...
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$99$th percentile of a sum of two random variables
Let $0 < p < 100$. The $p$th percentile of a random variable $X$ is the value $x_p$
which separates the smallest $p\%$ of the values of $X$ from the largest $(100-p)\%$.
In probabilistic terms, ...
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Restricted isometry property inequality
For a given integer $s \in \{1, \ldots, d\}$, we say that $\mathbf{X} \in \mathbb{R}^{n \times d}$ satisfies a restricted isometry property of order $s$ with constant $\delta_s(\mathbf{X}) > 0$ if
$...
- 2,316
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Pivot From a (Sufficient) Statistic
Asking for help in approaching a question from a Statistics textbook:
Let $X_1, X_2, ..., X_n$ independent and identically distributed with
density function $f_ {\theta}(x)$ and $T_n(X_1,X_2,...X_n)$ ...
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Can I assume an unbiased estimator of a differentiable statistic is also differentiable?
Let $F: \mathbb{R}^n \to \mathbb{R}$ be differentiable and $f: \mathbb{R}^k \to \mathbb{R}$ such that
\begin{align*}
F(x_1, \ldots, x_n)
&= \frac{1}{n!} \sum_{\sigma \in S_n} f(x_{\sigma(1)}, \...
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Concentration for Markov chain with spectral gap
Sub-Gaussian concentration for reversible Markov chains with spectral gap
Setup.
Let $(X_i)_{i\ge1}$ be a stationary, $\pi$-reversible Markov chain on a measurable space with spectral gap $\gamma>0$...
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The method of moments and maximum likelihood estimator for exponential distribution $\lambda^2 + \lambda$
Let $X_1,...,X_n$ be a random sample with exponential distribution Exp$(\lambda^2+\lambda)$
What is the method of moments estimator of $\lambda$?
So PDF is $F(x;\lambda) = (\lambda^2+\lambda)\exp(-(\...
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Why does the Brownian Motion not follow the Law of Large Numbers?
I am studying physics in third semester, and when I learned about the Brownian motion, I stumbled upon this counter-intuitive conclusion. Let me elaborate:
Assume Brownian motion in 1D, which can be ...
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Is there a way to simplify this expression for the adjusted Fisher-Pearson standardized moment coefficient
I managed to derive a simple incremental method to calculate the adjusted Fisher-Pearson standardized moment coefficient for a rolling window (see equation $[1b]$ in this article). More details about ...
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Uniform convergence in the central limit theorem
I was reading the following notes, https://www.cs.toronto.edu/~yuvalf/CLT.pdf, on the central limit theorem. I am a little confused about what the author says on page two,
"The exact form of ...
- 3,355
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analysis of the difference in the description of statistical distributions obtained mathematically/physically/computer-generated?
I plot the Gaussian distribution based on the mathematical definition and using the np.random.normal generator:
Next, using different interval steps and other parameters, I subtracted both twice to ...
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Tail bound of Gaussian likelihood ratio
Consider a hypothesis testing problem where X,Y are both $N(0,1)$ variables and under $H_0$, X,Y are independent and under $H_1$ (X,Y) follows a bivariate Gaussian with correlation coefficient $\rho$. ...
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Limiting Distribution of the MLE for a restricted Normal distribution
Problem: Let $X_1, \dots, X_n$ be iid drawn from the family of $N(\mu, \sigma^2)$ where we restrict $\mu \geq 0$. We'd like to find the limiting distribution of the mle of $\mu$.
It can be shown, and ...
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Looking for guidance: academic career prospects [closed]
I’m writing here because I could really use some advice. I got my bachelor’s and master’s degrees in math from a small university in the EU. Following the advice of some good professors, I came to UC ...
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Mathematically, why is statistical model is defined via multiple measures but not via multiple $\sigma$-subalgebras?
Mathematically, a statistical model is $(\Omega,\mathcal F,\{P_\theta\}_\theta)$ where $(\Omega,\mathcal F)$ is a fixed measurable space and $\{P_\theta\}_\theta$ is a family of probability measures.
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Hoeffding bound for random matrices proof question
The following is from High-Dimensional Statistics: A Non-Asymptotic Viewpoint by Wainwright.
Throughout, all matrices will be symmetric in $\mathbb{R}^{d \times d}$. For a matrix, let $\lVert A \...
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Confidence Interval for Reliability Weighted Samples
I'm trying to do statistical inference on a home poker game. I have calculated the winnings per hour, and I want to create a confidence interval for the variable winnings per hour, in say dollars.
The ...
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Complete description of the statistical properties of random functions [closed]
I'm working on a problem where I'm generating a smooth, periodic function of an independent variable, where this function is also a function of a number of random variables. Thus the function itself ...
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pdf of sum of two uniform distributions
Let $X_1, X_2$ follow the uniform distribution of U(0,1), and $X_1, X_2$ are independent.
Also Let Y be sum of $X_1, X_2$, that is $Y=X_1+X_2$,
To calculate the pdf of Y, I used the cdf technique.
I'...
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Does the union of two datasets form a mixture distribution? [closed]
I have two datasets: $A := \{X_i\}_{i=1}^{n_a}$ sampled from distribution $P_A$, and $B := \{X_j\}_{j=1}^{n_b}$ sampled from distribution $P_B$.
Let $n = n_a + n_b$ be the total sample size, and ...
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definition of estimator in random variable estimation
I am studying random signals and noise (a course for EE students, but mathematical and formal), and have a question about the definition of an estimator (in the context of estimating a random variable ...
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Generalization of Lindeberg's condition for sequences of finite sequences of random variables
For each $d \in \mathbb{N}$, I have a finite sequence of independent real-valued random variables $\{X_{d,k}\}_{k \in [d]}$, but the marginal distributions change for different choices of $d$, i.e., $\...
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What exactly is a distribution function? [closed]
I’ve read about the binomial distribution, Poisson distribution, geometric distribution, hypergeometric distribution, and negative binomial distribution. I understand their formulas and can solve ...
- 31
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Symmetric multivariate Laplace distribution has infinite density at the mean?
I was trying to implement the multivariate generalisation of the Laplace distribution described in the paper by Eltoft et al. 10.1109/LSP.2006.870353 . In it the authors derive the $d$-dimensional ...
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Delta method measurbility question.
Suppose we have a random variable $\hat \theta_n$ such that $\hat \theta_n \to \theta_0$ in probability. Let $f \colon \mathbb{R} \to \mathbb{R}$ be infinitely differentiable function. Then, the delta ...
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Uniqueness of the function $\tau(\theta)$ in the Cramér-Rao inequality
Theorem (Cramér-Rao inequality).
Consider a sample from a parametric model satisfying regularity conditions.
Let $\theta^*$ be an unbiased estimator of $\tau(\theta)$. Then for any $\theta \in \Theta$,...
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Why is X following $\mathcal{N}(\mu + \Lambda z, \Phi)$ in the Factor Analysis model?
I’m working through some notes on Factor Analysis and I noticed something that confused me.
We have
$ X = \mu + \Lambda z + \epsilon $
with
$z \sim \mathcal{N}(0,I_s)$,
$\epsilon \sim \mathcal{N}(0,\...
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Optimal draft position in a 12 man fantasy football snake draft
I’m trying to analyze fantasy football drafts from a mathematical/statistical perspective, specifically for a 12-team snake draft. I want to determine which draft position is “best” in terms of ...
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