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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$X-Y$ equivalent in distribution to $0$?

If $X$ is equal to $Y$ in distribution, is it equivalent to $X-Y$ which is equivalent in distribution to $0$?
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The Birthday Problem

I've been reading about the birthday problem which, as I'm sure many of you will know, is a statistical problem which aims at finding out the how many people you would need in a random group to be ...
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Highest points in heart cycle graph

I'm making an application that reads the heart cycle from a device, and I've aimed to get this image: Now, I need to get the highest points that appear in every cycle in order to calculate the ...
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281 views

Central Limit Theorem Definition

My friend and I have a bet going about the definition of the Central Limit Theorem. If we define an example as a number drawn at random from some probability density function where the function has a ...
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179 views

Conditioning on an event with probability close to one

Let $(\Omega,\mathcal{F},P)$ be a probability space. If $A\in\cal F$ is an event with $P(A)=1$, then $$ P_{\mid A}(B)=P(B\mid A)=\frac{P(B\cap A)}{P(A)}=P(B),\quad B\in\cal F. $$ I wonder if something ...
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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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145 views

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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159 views

Is this function concave or can it be made concave?

I am working with a point process with an event arrival rate of: $$ \lambda(t) = \mu + \sum\limits_{t_i<t}{\alpha e^{-\beta(t-t_i)}}$$ where $ t_1,..t_n $ are the event arrival times. The log ...
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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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190 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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252 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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119 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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231 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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154 views

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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615 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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424 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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374 views

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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241 views

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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125 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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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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695 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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72 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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520 views

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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505 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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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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276 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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Finding an upper bound for $\frac{d}{d\theta}\beta^*(\theta)|_{\theta=\theta_0}$

Suppose that a random variable X has a distribution depending on a parameter $\theta$, $\theta \in \Theta$, and consider a test of hypothesis $H_0: \theta = \theta_0$ versus the alternative $H_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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47 views

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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127 views

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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215 views

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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473 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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996 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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84 views

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} ...