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.

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Curve Fitting Problem

Find best values of a and b if y = ax + b log x is the curve which represents most closely the observed values given below : X: 2, 3, 4, 5, 6 Y: 3.74, 5.99, 7.47, 8.92, 9.86
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How to show the expectation of gradient estimate is unbiased for least square problem?

Suppose we are given $m$ points $\{a_1, \cdots, a_m\}$ where $a_i \in \mathbb{R}^n$, and the corresponding target values are $\{b_1, \cdots, b_m\}$ where $b_j \in \mathbb{R}$, and we want to solve the ...
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Testing significance of patterns of results

I'm a high school English teacher conducting an independent study, and I'm a total novice to statistical analysis, so please forgive me if I mischaracterize anything. I have gathered pretest and ...
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Implication of Chebyshev's inequality

Is it correct that Chebyshev's inequality for a random variable $X$ implies that \begin{align*} P(|X|>\delta) \leq \frac{E[X^k]}{\delta^k} \quad \text{for } k=2,4,\dots \text{ and } \delta >0 \...
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is crosscorrelation function with 0 lag the same as correlation between the change at each time step?

If I have two time series, $\vec{x} = [x(t_1), x(t_2), \dots]$ and $\vec{y} = [y(t_1), y(t_2), \dots]$, and I compute the crosscorrelation function on these two vectors with a time lag of $0$, is this ...
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I have a jar with N unique balls. If I draw N random balls with replacement, what is the most likely number of duplicate draws?

I have a jar with N unique balls. If I draw out N random balls with replacement, how many duplicate draws am I likely to have seen when I'm done? I.e., what is the most likely number of times that I ...
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Law of Kolmogorov-Smirnov statistic

Do you have a easy proof of the fast that the Kolmogorov-Smirnov statistic law is continuous ? Because I don't know anything about Brownian motion. The statistic is defined here. Thanks and regards....
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Mixture prior of waiting times

Consider there are two equally trustworthy experts, each of them has a belief of the waiting times of a event. Expert A says it's between 5-10 minutes while B says 0-25 minutes. I'm wondering how to ...
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Developing a model to approximate an unknown score from known scores

Apologies if this would be better suited elsewhere, this is the SE I was recommended. I can delete / move if needed. The target is to approximate a hidden rank in a game (Dota 2 if anyone is ...
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Stuck on probability/statistics question

EDIT: Post as been edited to address relevant questions raised in comments. I'm new to the site and I'm stuck on a probability question. I don't think it's trivial, certainly not to me, as I am ...
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Problem Interpreting histogram

I am reading Understandable statistics. In the second chapter, example 2, the author shows the following histogram: The author then proceed to interpret the histogram as follows: About 40% of the ...
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Does Causation imply dependency in the Euler scheme discretization for SDEs?

My question may sound stupid but I have a confusion concerning the following statement: Assume we have n-observations $(X_0,\dots,X_n)$ that are coming from a discretization of an SDE using the ...
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Accuracy of estimation of the variance

Consider $N$ i.i.d. samples $x_1, \dots, x_N$ from an unknown discrete distribution on $\{0,\dots, n\}$. We know that $$\frac{1}{N-1} \sum_{i = 1}^N (x_i - m)^2$$ where $m$ is the sample mean, is an ...
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generalized bayes rule?

The Bayes Rule, as I learned, is given by: $$P(\theta|X) \propto P(X|\Theta) \times P(\theta). $$ But I'm reading a paper, which applies Bayes rule in a weird way: $$P(\theta|X,Y,Z) \propto P(X,Y|\...
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2 Sample T-Test vs 1-Factor ANOVA

Is it more advantageous to conduct a 2-sample T-test or single factor ANOVA when determining statistical significance between the means of two independent groups? I have run sample tests on mock data ...
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Probability of observing at least $k$ heads if coins have different probabilities of heads? [duplicate]

I have a collection of Bernoulli RVs $\{X_1, \ldots, X_N\}$ and know the success probabilities of each: $\{p_1, \ldots, p_N\}$. Is there, in general, a way to efficiently compute the probability of ...
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Fisher's exact test two sided confusion

Consider the guessing milk tea example $$\begin{array}{c|c|c|} & \text{Guess Milk} & \text{Guess Tea} \\ \hline \text{Milk} & 3 & 1 \\ \hline \text{Tea} & 1 & 3 \\ \hline \...
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Infinite monkey theorem independent of number of monkeys

I was just thinking about the infinite monkey theorem and arrived at a creepy conclussion. Let's begin with the probability that one monkey types a set of $N$ letters with a chosen order, which is ...
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Linear regression for gaussian [closed]

Consider the following regular model $$ Y_{i,j} = a + bx_{i,j} + \epsilon_{i,j}, 1 \le i \le p, 1 \le j \le n_{i} $$ Can you test the hypothesis, the linear regression (line) pass by the point $(x_{...
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GLRT statistic for composite normal hypothesis, two unknowns

Suppose $X_1...X_n$ ~ iid ~ $\mathcal N(\mu, \sigma)$, both parameters unknown. We want to test $H_0: \mu \leq \mu_0, H_1: \mu > \mu_0$. Show that the LRT (likelihood ratio test) statistic is ...
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Order Statistics and Convergence in Probability

Let $X_1, ..., X_n$ be iid continuous random variables with cdf $F$ and define $Y_n(x) \equiv \sum_{i=1}^n 1(X_i \leq x)$ Define the inverse cdf as $F^{-1}(y) = \text{inf}\{x \in \mathbb{R} : F(x) \...
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Question on reverse union bound for independent events

From the note on union bound I found the following Fact 1.3 (Reverse Union Bound) Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be arbitrary events, not necessarily independent. Suppose that $\...
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Can a small sample size with applied machine learning provide higher accuracy than the sample size alone?

Say I have 4000 apples and I want to know the percentage of apples that are red and the percentage that are green. Using only a sample, I could randomly pick and classify ~351 apples to report with a ...
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Continuity Correction for Random-Walk

If I've been asked to use a normal approximation for estimating the probabilities of a simple random walk being within two values, knowing the the simple random walk can only result in even numbered ...
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What is the definition of 1 standard deviation.

I know what the standard deviation is. I know about the 68–95–99.7. My question is what is the definition of 1 (just 1) standard deviation. Why is it 68 and not 70. Is it because that's where the ...
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Why must a manifold be orientable to be able to induce a specific statistical manifold?

Let $M$ be a manifold, $g$ a Riemannian metric, $\nabla$ be an affine connection on $M$, and $\nabla^{*}$ the unique dual affine connection of $\nabla$ on $M$, i.e. for all vector fields $X,Y,Z$ on $M$...
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how to show sufficient statistic

Let $X_1, X_2,... X_n$ denote a random sample from a geometric distribution with the parameter $\theta$. Show that $\sum_{i=1}^n X_i$ is a sufficient statistic for $\theta$. I know that a statistic ...
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Confidence bounds for nonuniformly distributed 2D data

For a set of measurements I’d like to know how accurate my results (mean values) are. My dataset (example) consists of a number of $(x_i,y_i)$ values (The total number $i$ of data points depends on ...
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Problem involving a probability function that consists two functions, and the input to each function.

I was trying to solve this problem: $f(x) = k[\phi (x) + \lambda g(X)]$ Where: $\phi(x)$ is a normal distribution with mean =0 and variance =1. $g(x)$ is defined as: $\frac{1}{\lambda}$ for $ \leq x \...
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Standard Error and Standard Deviations

How it s possible to calculate standard error for single sample, While standard error is defined as variance in different sample means?
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Standard Error calculation [closed]

How many standard errors around the hypothesized value should we use to be 98 percent certain that we accept the hypothesis when it is true?
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Likelihood Ratio Test for the Normal Distribution with unknown mean

Let $(X_{1},...,X_{n})$ a $n$ sample of the law $N( \mu, \sigma^{2})$. We assumed we don't know $\mu$ and $\sigma^{2}$. Let $\mu_{0} \in \mathbb{R}$. Show that the Likelihood-ratio test for $\mu =...
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Logit regression change in $x$

I've been reviewing the logit regression equation to predict the probability of a response variable given $X=x$ as per below and I can't understand how a per unit change in the predictor is equal to $...
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AP Stats questions on significance tests

A company produces millions of 1-pound packages of bacon every week. Company specifications allow for no more than 3 percent of the 1-pound packages to be underweight. To investigate compliance with ...
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UMVUE for a $NB(k,p)$ with $g(p)=p^k$

I am prepping for an exam and this is one of the questions the professor handed to help us prepare. $$X\sim NB(k,p)$$ where $k$ is the number of successes. Find the UMVUE of $g(p)=p^k$ if it exists....
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Create a probability model from discrete data

I have a set of data about how long a task takes that I expect to be normally distributed (fits a bell curve). I would like to create a curve/distribution that fits this data as best possible so that ...
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Expectation of the maximum eigenvalue.

Let $X_1, X_2, \ldots X_n$ identically distributed independent random variables with zero mean and finite variance $\sigma^2$. Let $\bar X$ be the sample mean and consider the random vector $ B= n^{-1/...
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Using convolution formula to find PMF and then to show negative binomial distribution

Let $X$ and $Y$ be independent random variables taking only integer values. Let $Z=X+Y$, which also takes only integer values. Its PMF can be computed by the convolution formula: for any integer $z$, ...
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How do you calculate the mean value for a percentage chance? For example: Damage per second with an item that has crit chance?

To elaborate on the example: if I have a weapon that does 6 damage, and attacks 1 time per second the calculation is damage * attacks per second. But what if the weapon has a 5% chance to strike ...
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conditional expectation and variance in regression

Given: \begin{aligned} X \in \mathbb{R}\ X \sim N(\mu_x, \lambda_0^{-1}). \\ Y_1,...,Y_n \in \mathbb{R} \mbox{ are all independent and distributed each with} \\ N(x,\lambda^{-1}) \mbox{ given some }x....
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Decision Tree Problem: Evaluate probabilities and determine in terms of C, all the optimal decisions.

I'm struggling with this decision tree question: A part of an aircraft engine can be given a test before installation. The test has only a 75 % chance of revealing a defect if it is present, and the ...
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Confidence interval obtained by Hoeffding's inequality is the "worst-case''?

Hoeffding's inequality can be used to generate a confidence interval. I wonder if Hoeffding-inequality is the "worst-case" approach (among all other common approaches in textbooks, e.g. assuming the ...
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Confidence Intervals (A level)

Firstly let me apologise for asking this question on here - I see this as getting a sledgehammer to crack a nut, but I have nobody else that I can ask for advice on this topic. I'm an A level ...
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Counter intuitive Prior/Posterior relationship in Bayesian inference for estimated probability fusion

I am trying to infer the probability distribution of a binary variable $X$ (True or False) using observations $O = \langle O_1,O_2,\ldots,O_n\rangle$, mutually independent given X. I also have a ML ...
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Terminology: Average vs Statistic

When is it correct to call a calculation an average and when is it correct to call a calculation a statistic? I recall reading something along the lines of: an average is calculated directly from a ...
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Mahalanobis distance : invariant by change of coordinate system

My point is the following: I think that I should obviously find the same distance of Mahalanobis between two points x and y according to whether I calculate this distance on my starting space or on ...
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Dependence of variance estimate regarding periodic or nonperiodic system

It has a physical background, but is more kind of a statistical question, so I thought it would fit here very well: For my master's thesis in biophysics I'm studying discrete one-dimensional systems. ...
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Variance of the weighted sample standard deviation

According to this Mathworld page, the variance of the sample standard deviation $s$ can be analytically computed with eq. (10), assuming a normal distribution. $$ \mathrm{var}(s) = \frac{1}{N} \bigg[...

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