Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis.

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Probability/Combinatorics Problem. A closet containing n pairs of shoes.

A closet contains n pairs of shoes. If 2r shoes are chosen at random, (where 2r < n), what is the probability that the chosen shoes will contain no matching pair? I have tried thinking about this ...
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Greatest common denominator of measurements

In a couple months, I'll do the Millikan experiment. Then, I'll end up with a number of charge measurements and their errors $$((q_i, \Delta q_i))_{i \in \mathbb N}.$$ The idea is that all those $q_i$ ...
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Proof of $ \text{Var}\,\left(\sum_{i=1}^{n}g(X_i)\right)=n\left(\text{Var}\,g(X_1)\right).$

I have a question about part of a proof of a Lemma in a book (Casella's Statistical Inference) I'm reading. This it how it goes. Let $X_1, \cdots ,X_n$ are a random sample from a population and ...
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188 views

Finding the joint distribution of $X_{1:n}$ and $\overline{X}$

I need to show that, given a random sample of independent variables $X_1, ... , X_n$, each following a distribution EXP($\theta$,$\eta$), that is, ...
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What can we conclude from correlation?

I just got my statistics test back and I am totally confused about one of the questions! A study was done that took a simple random sample of 40 people and measured whether the subjects were ...
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115 views

Expected overlap

Suppose I have an interval of length $x$ and I want to drop $n$ sticks of unit length onto it (where $\sqrt x<n<x$). What is the expected overlap between sticks? ($x$ can be assumed to be large ...
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181 views

Jensen's inequality

I am using Jensen's inequality and conditional expectation to prove the following inequality: Let $\lambda_i$ be real for $i\in \{1,2,...,M\}$ and $\bar{\lambda}=\frac{\sum_{i=1}^M\lambda_i}{M}$. ...
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234 views

Statistics formula for wifi positioning.

Assuming I have $3$ access point namely: $AC_1$, $AC_2$ and $AC_3$ and I want to know my location using this access point and a mobile device that will get signal from the access points. First thing ...
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117 views

Calculating the average of a possibly infinite “compound” length

Sorry for the ambiguous title, I couldn't find a good word to describe my problem. So here is my problem: You are a player, and you have a dice. You have N number of throws available then you can't ...
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220 views

Is there an introduction to probability and statistics that balances frequentist and bayesian views?

Perhaps, roughly, I might be described as advanced undergraduate regarding mathematics. However, I have not learned statistics and have only learned elementary probability. Does there exist a book or ...
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Disprove independence of vector of Gaussians by independence of marginals

If we have three random variables $X,Y,Z$, then if $X$ and $Z$ are independent, and $Y$ and $Z$ are independent, it doesn't follow that $Z$ is independent of the vector $(X,Y)$. There is a simple ...
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Why $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$

I'm reviewing probability and statistics.The textbooks said that if the sampled population is infinite, then $$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$$ I'm curious about how does this ...
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594 views

Eigenvalue decomposition of block covariance matrix for Canonical Correlation Analysis (CCA)

Edited: My question is related to a tutorial I was reading. The covariance matrix is a block matrix where $C_{xx}$ and $C_{yy}$ are within-set covariance matrices and $C_{xy} = C_{yx}^T$ are ...
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420 views

Expectation and Variance of Ratio Estimator

Let $X$ and $Y$ be positive random variables such that $$E(Y\mid X)= aX $$ $$\operatorname{Var}(Y\mid X) = b^2X^2 $$ $$a,b > 0 \text{ are constants}.$$ Let $R = ...
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Help with question on joint Gaussian distribution

Does anyone know how to start this question? Let random vectors $x,u,v$ have joint Gaussian distribution, and $u,v$ be independent. Show that $E(x|u,v)=E(x|u)+E(x|v)-E(x)$.
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Can I build a program that will tell me if a real world data set looks linear, logarithmic, exponential etc?

I have a bunch of real world data sets and from manually plotting some of the data in graphs, I've discovered some data sets look pretty much logarithmic and some look linear, or exponential (and some ...
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Update a regression on the fly?

Say I have 100 people each with a height, weight, and age. I make a regression that predicts age based on height and weight. Now, I would like to update that model when I meet someone new. I don't ...
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122 views

Density function of $\max(X_1,\dots,X_n)$.

I'm making this statistics exercise and I'm not sure about my solution. Find the density function of $Y=\max(X_1,\dots,X_n)$ if they are all i.i.d. This was my take on this question: $F_Y(a)=P(X_1 ...
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110 views

Calculating success chance from algorithm

Not super sure this is the right *exchange for this question, but here we go. Let's say I'm writing a game, and in this game the player may attack another unit. The chance of hitting is an "opposed ...
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227 views

Working out minimum sample size

I have just started a course in statistics and have some general questions that have arisen trying to solve the following question: A survey organisation wants to take a simple random sample in order ...
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691 views

Finding the MLE for parameter $\theta$ from distribution of the form $e^{-|x-\theta|}$

this is my first post so I apologize if the formatting is a little rocky. I'm currently going through "Probability and Statistics" 4th ed by DeGroot/Schervish, and I was wondering if somebody could ...
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71 views

Drawing previously undrawn cards from a deck

Suppose you have a deck of $y$ cards. First, randomly select $y-x$ distinct cards and sign the face of each, then shuffle all the cards back in to the deck. Proceed as follows: Draw a card. If it is ...
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96 views

How to prove that second derivative of $\log\big(\int_{-\infty}^x e^{\frac{-t^2}2} dt\big)$ is $>-1$?

Let $\Phi(x)=\int_{-\infty}^x e^{\frac{-t^2}2} dt$. How can I prove that $$\left[\frac{e^{\frac{-x^2}2}}{\Phi(x)}\right]'>-1?$$ I could prove that its $lim$ at $-\infty$ is $-1$ and at $\infty$ it ...
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Median of order statistics

I recently learned that to find the pdf of the median of say $X_1,X_2, X_3$, you first find the Cdf via $$ P(M \le x) =P(\text{at least 2 are}\, \le x) = P( \text{exactly 2 are}\, \le x) + P(\text{all ...
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498 views

Calculation of the moments using Hypergeometric distribution

Let vector $a\in 2n $ is such that first $l$ of its coordinates are $1$ and the rest are $0$ ($a=(1,\ldots, 1,0, \ldots, 0)$). Let $\pi$ be $k$-th permutation of set $\{1, \ldots, 2n\}$. Define ...
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136 views

Poisson Process - Courts

IITK sports facility has $4$ tennis courts. Players arrive at the courts at a Poisson rate of one pair per $10$ min and use a court for an exponentially distributed time with mean $40$ min. Suppose ...
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150 views

Show that the posterior density of ($\mu$, $\tau$) is equal to $f(\mu, \tau | x_1, …, x_n) = f(\mu| \tau, x_1,…,x_n)f(\tau|x_1,…x_n)$

Here is the full problem: Let $X_1,...,X_n$ be a random sample from a $N(\mu,\sigma^2)$ distribution. Let $\tau = \sigma^{-2}$, so we can write the distribution as $N(\mu,\tau^{-1})$. Suppose the ...
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What's the expected value of average absolute deviation from the mean of k randomly picked numbers?

Say we have to randomly pick k integral numbers out of n. The numbers are from the range < a; b >. What is the expected value of average absolute deviation from the mean for that random subset of ...
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71 views

Getting a single-value estimation of trust in a computed mean

Suppose I have a number N of independent ratings of a given item, where each rating is an integer between 1 and 7 (inclusive). For simplicity sake, let us assume the ratings are normally distributed, ...
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274 views

expected value for min estimated entropy

We have a random generator that generates independent random bits with probability $P(x=1) = P$ and $P(x=0)=1-P$. Given $N$ random independent bits, we estimate $P$ by $\hat{P} = N_1/(N_0+N_1)$. ...
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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 & ...
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Distribution for ratio of dependent quadratic forms.

Random vector $\mathbf{x}_{0}$ $\sim$ $\mathcal{N}\left(\boldsymbol{\mu}, \mathbf{\Sigma} \right)$ is a sum of two orthogonal random vectors: $\mathbf{x}_{0}$ = $\mathbf{x}_{1}$ + ...
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Random point distribtion

How to generate numerically a set of random points $(x_1,y_1), (x_2,y_2),\cdots, (x_N,y_N)$ such that the pair-wise distances $d = \sqrt { (x_i-x_j)^2 + (y_i-y_j)^2}$, for all $ 0<i\le N, ...
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Deconvolution of distribution of diffraction reflexes

I'm a chemist stuck in a mathematical problem. Please bear with me as I'm trying to express myself in Math language. Let me explain in short terms the experimental method I'm using: X-ray ...
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Clarification in a paper

This is regarding a clarification in page 384 of a paper published in Annals of Statistics by Amari. In page no. 384, he defines $$R_i(t)=\frac{\partial}{\partial \theta_i} ...
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Multivariate Gaussian equivalent for a Gaussian integration identity.

For a one-dimensional x, $$\int_{-\infty}^{\infty}x^{2}e^{-x^{2}}dx=\frac{1}{2}\int_{-\infty}^{\infty}e^{-x^{2}}dx$$ This can be shown through integration by parts. There is a good derivation of ...
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Is there a way to exploit the fact that the covariance matrix has a blocked structure to more easily compute the multivariate normal density?

I'm trying to minimize the (negative) multivariate normal log likelihood (dropping constants): $$ \log |\boldsymbol\Sigma|\,+(\mathbf{x}-\boldsymbol\mu)^{\rm ...
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Gaussian Mixture Model

I am fitting a Gaussian Mixture Model to high-dimensional data (40 dimensions) I have trained the model using EM, learned the parameters and now I want to know quantitatively what is most important in ...
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462 views

Idempotence and the Rao–Blackwell theorem

Original question: In the Wikipedia article on the Rao–Blackwell theorem, we read: In case the sufficient statistic is also a complete statistic, i.e., one which "admits no unbiased ...
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Correlation Coefficient and Determination Coefficient

I'm really new to linear regression and am trying to teach myself. In my textbook there's a problem that asks why R^2 in the regression of Y on X = the sample correlation between X and Y the whole ...
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976 views

Standard Deviation: Why divide by $(N-1)$ rather than $N$?

The forumlae for standard deviation seems to be the square root of the sum of the squared deviation from mean divided by $N-1$. Why isn't it simply the square root of the mean of the squared ...
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A maximal Hoeffding's inequality?

Let $X_1, \cdots, X_n$ be real-valued independent random variables satisfying $|X_k|\le 1$ and $\mathbb EX_k=0$. Hoeffding's inequality tells us that for any $k=1,\cdots, n$ and $t>0$, $$\mathbb ...
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Sum of two independent geometric random variables

Let X and Y be independent random variables, $ P(X = k) = P(Y = k) = p(1 - p)^{k-1} $ How do you show that the pmf of $ Z = X + Y $, is negative binomial, and how do you find $ P(X = Y) $?
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Integral with Normal Distributions

I know that the following equality is true for any $a$ and $\sigma$ (I have solved it numerically): $$\int_{-\infty}^{+\infty}\Phi\left(\frac{a-x}{\sigma}\right)\frac1{\sigma} ...
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What does it mean to integrate with respect to the distribution function?

If $f(x)$ is a density function and $F(x)$ is a distribution function of a random variable $X$ then I understand that the expectation of x is often written as: $$E(X) = \int x f(x) dx$$ where the ...
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simulating a fair random process with an unfair one.

Let's say I have a stochastic process that outputs $1$ or $0$ with probability $p$ or $1-p$ respectively, $p\neq 1/2$. Let's assume this is a repeatable iid process. So I can generate $X_1,X_2\dots$ ...
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Why do Mean, Median, Mode, and Range present in school lessons?

I studied in East Europe and post Soviet mathematical education program have no Median, Mode, and Range terms. Mean (or average) on other hand was studied (with root mean square and sometimes with ...
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What does expected value of sum of two discrete random variables mean?

I am confused with summing two random variables. Suppose $X$ and $Y$ are two random variables denoting how much is gained from each two games. If two games are played together, we can gain $E[X] + ...
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P(You have the disease | The test says you do)

You are diagnosed with an uncommon disease. You know that there is only a 1% chance of getting it. Use the letter D for the event "you have the disease" and T for "the test says so." It is known that ...
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Optimal number of answers for a test with wrong-answer penalty

Suppose you have to take a test with ten questions, each with four different options (no multiple answers), and a wrong-answer penalty of half a correct answer. Blank questions do not score neither ...