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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Proof that data-processing selection process is statistically independent of the sample?

I hope to replace a proof of my own, in a paper explaining that the "no free lunch" theorems for optimization actually address sampling and statistics, with a reference to an existing result on ...
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23 views

Calculating close to a uniform distribution where the mean is different then the midpoint?

I need some help understanding what type of distribution calculation I need. I am attempting to achieve as close to a uniform distribution as possible, but subject to the constraint that the mean of ...
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41 views

Two stage GMM estimator in Matlab

I am trying to create a simple GMM estimator for the mean of a normally distributed random variable using the first three odd central moments of a normal distribution (all of which should be zero ...
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21 views

Check work for finding Max log-Likelihood of a geometric Distribution

Here is my geometric distribution: $P(L=n)=p^{n}*(1-p)$ To find the max likelihood, I do: $\sum_{L_i} L_i\log(p) + \log(1-p)$, where L_i is a particular length. I take the derivatives and end up ...
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37 views

Notation for parametric family of joint densities

This is copied from my textbook; Assume that the statistical model for the MVR $\textbf{Y}=(Y_{1},Y_{2},\ldots , Y_{n})^{T}$ is given by the parametric family of joint densities: $$\{ ...
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195 views

Measure dispersion of a set of values resides in a range

I want to know what is best method to calculate (measure) the statistical dispersion of a set of values resides between a range. Scenario: My goal is to build an index. I have two methods that ...
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1answer
81 views

probability of bingo

It is the first time I heard about bingo game and I would like to learn more on this game by mathematical analysis. To make it simple, I consider the American BINGO with 75 balls used and each game ...
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36 views

The use of Weighted arithmetic mean to determine the extent to which the data correspond

I don't know when I can ask such question, because it is not a completely mathematic question. I have a text and a topic (single word). I want to check how much this text corresponds to the topic. ...
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1answer
49 views

Number of channels required to provide access to subscribers 80% of the time

A multichannel microwave link is to provide telephone communicationto a remote community having 12 subscribers, each of whom uses the link 20% of the time during peak hours. How many channels are ...
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32 views

Hypothesis Testing for Distributions

I am working on a practice prelim question. I just need to describe a best test of hypothesis for testing the probability density function of $$ H_0: f(x)= f_1(x)$$ vs. $$ H_1: f(x) = f_2(x) $$ ...
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2answers
33 views

Does the parameter change during data generation in Bayesian Inference?

Let's assume that we have the following graphical model: This graph encodes the joint distribution $P(p,x_1,x_2,x_3,x_4) = P(p)\prod_{i=1}^{4}P(x_i|p)$. In the Bayesian inference, if we know ...
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1answer
65 views

normality of data

Does the qqplot below suggest that the data is normally distributed? The fact that it's nearly perfectly linear is to me an indication of normality. However, the Anderson-Darling test for some reason ...
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81 views

Uniform sampling with replacement item frequency

Suppose we are sampling from $N$ distinct items uniformly with replacement $M$ times. What can be said about the distribution of frequencies of items drawn? For example, if I sort all the frequencies ...
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31 views

Calculate P value from Z value

My data consists of values for 192 countries. I want to calculate the outlier value. For this I first calculated the z values for each using the formula for Z scores. Now I want to calculate the P ...
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100 views

Statistical symbols: should greek letters be used for population or for a sample?

When finding mean and the stansard deviation, do you use the Greek symbols for a population, or a sample. When do you use "s" and xbar, for a population, or a sample? (I am taking AP Statistics)
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33 views

proof regarding kurtosis

I am trying to manipulate an expression about kurtosis. It says that $kurt(X)=\frac{V_4(X)}{\sigma^4}-3$, $V_4(X)=E[(X-\mu)^4]$ Then it states that $kurt(\bar{X})=\frac{k}{n}$, where $k = ...
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1answer
59 views

Question about the Bayesian Inference of a parameter

In order to understand the difference between the Frequentist and Bayesian inference, I was reading the presentation at: http://www.stat.ufl.edu/archived/casella/Talks/BayesRefresher.pdf . In order to ...
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75 views

How does research in math differ from research in statistics?

I'm at a crossroads where I'm considering switching my major from electrical engineering to math, because quite frankly, I'm just not getting enough math to satisfy my passion from engineering. While ...
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163 views

Jacobian for a matrix transformation: Example of Cholesky decomposition

I would like to generally understand how the Jacobian of a matrix transformation can be computed. As a concrete example, consider the Transformation from a (correlation) matrix to its Cholesky ...
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1answer
34 views

Statistics; looking for a practical example-based book

I'm looking for an example-based 2nd year or thereabouts undergraduate statistics textbook in the style of Engineering Maths by Ken Stroud ...
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1answer
50 views

Hypothesis testing. Help verify and interpret the solution.

This is a question from Statistical Theory I have encountered. I have almost solved it, but have some trouble interpreting the solution. Something seems weird, and I am not sure whether I am entirely ...
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19 views

factor models and using cross sectional regression?

I have been doing some reading on factor models. In the literature it mentions that when creating a portfolio that maximises particular attributes it may lead to unwanted bias to other factors. I ...
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1answer
22 views

Given gender distribution of two programmes, which one does class belong to?

Classes at a high school are divided into two branches: Mathematics (65% boys) Languages (45% boys) Matthew observes that 55% of his class is made up of boys. Which branch does it most likely ...
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36 views

How to Find Symmetric Equilibrium Bidding Strategy in Mixed Auction?

N-bidders take part in a mixed auction (mix of first- and second-price auctions). The highest bidder (bidder 'i') wins and pays a*b_i + (1-a)*(max b_j) ('i' is not equal to 'j'), where 'a' is fixed ...
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1answer
40 views

Uniqueness of solution to quantile minimization problem

I read here: http://librarum.org/book/11685/31 (p. 51, Ex. 3) that quantiles are solutions to certain minimization problem. Here is the proof: ...
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1answer
58 views

Specify the distribution of two discrete independent variables

So I'm preparing for a reexamination for an introductory statistics course, last time I had trouble finding a way to specify the distribution of variables/vectors. So my question is, maybe a more ...
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65 views

Prove Logarithmic function is part of exponential family

The aim is to prove that the logarithmic distribution with parameter $p (0<p<1)$ is part of the exponential family and hence, give its canonical parameter. To prove a distribution is part of ...
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1answer
37 views

2-dimensional random walk, covariance

Let $X_1, \ldots X_n \sim N \left( \left[ \begin{array}{c} 0 \\ 0 \end{array} \right], \left[ \begin{array}{cc} 1 & \rho\\ \rho & 1\end{array} \right] \right), S_n = \sum \limits_{i = 1}^n ...
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3answers
126 views

“Practical” Claim about Hypothesis Testing of Bernoulli Distribution Parameter

First, let me state the original problem (in my own wording): Describe the decision procedure for testing the hypothesis about the parameter $p$ (success rate) of a Bernoulli distribution. The ...
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2answers
67 views

Optimize rate of collection in counters

Suppose you have $K$ counters. The value of these $K$ counters are all $0$. Every second, each counter has a $J$ chance of incrementing itself, up to a max value of $I$. Every second, you may choose ...
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1answer
33 views

Show that statistic is (not) sufficient

I need to verify ifthe statistic $|X|$ is or npt sufficient for $\mu$, if $ X \sim N(\mu, 1)$ Using the definition, I've obtained the pdf of X given $ T(X)=|X|:$ $$f_{X|T}(x|t) = ...
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2answers
42 views

Normal distribution and algebra problem

Bags of cement are labeled $25 \operatorname{kg}$. The bags are filled by machine and the actual weights are normally distributed with mean 26.0 kg and standard deviation $0.50 \operatorname{kg}$. It ...
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14 views

Efficiency of the sample variance for Student's t distributed observations

my question refers to the book of Harvey "Dynamic models for volatility and heavy tails". He stated (p. 20) that the efficiency of the sample variance for t-distributed observations is given by ...
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1answer
35 views

Given a set of partial orderings of samples from a set of distributions, can we estimate the (relative) mean of the distributions?

This problem is motivated by attempting to construct a total ordering out of an arbitrarily large set of potentially contradictory partial orderings. Let's assume we have some set of items I for ...
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1answer
35 views

How to get a function similar to the ELO ranking system

I'm not an expert in maths. I'm looking a function similar to the ELO system in order to give the users a score. Those are the requirements for the system: The score must be between -100 and 100 ...
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1answer
35 views

How to check hypothesis in statistical data?

I have a statistical problem. In a city there are some hostels which differ by the number of rooms. The input data are the following. In a table there is information about hostels and corresponding ...
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28 views

kullback liebler divergence for correlated processes

Suppose $X_n^{(1)}=\lambda_1 X_{n-1}^{(1)}+\mu_1+\epsilon_n^{(1)}$ and $X_{n}^{(2)}=\lambda_2X_{n-1}^{(2)}+\mu_2+\epsilon_n^{(2)}$ where $|\lambda_i|<1$ for $i=1,2$ and $\epsilon_n^{(i)}$ are ...
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1answer
14 views

What will be the pdf when Chi-square distribtion function is multiplying by variance

What will be the pdf of given function in which $\sum _{n=0}^{N-1}(x^2[n]/\sigma^2)$ is Chi-square distribtion function multiplying by variance $\sigma^2$ : $\sigma^2\sum ...
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3answers
473 views

Minimum number of attempts to guess a PIN code, given constraints

I'm playing a video game at the moment called Sleeping Dogs, in which some of the mini-missions are to 'hack' a security camera, by guessing a four-digit PIN code. Here are the rules: 1) You are ...
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3answers
50 views

Exact interpretration of p-value and significance of test

First question: Let's say we have a hypothesis test: ${ H }_{ 0 }:u=100$ and ${ H }_{ 1 }:u\neq 100$. The sample has a size of 10 and gives an average $u=103$ and a p-value = 0.08. The level of ...
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1answer
67 views

Calculating the chance of something happening over and over again

I'm trying to calculate the probability, and potential cost on society, of people returning to homelessness after going through the system one, two, or several more times. Let's say that someone who ...
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17 views

How to standardise with respect to two variables?

I'm programming an app and I have a resizeable zone and a map of police incidents and where they've occurred. I've trying to get a result between $0$ and $1$ (or $0$ and $100$) that is calculated with ...
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38 views

Is there a name for sum over one set divided by the cardinality of another set?

What is the summation of one set real numbers divided by the cardinality of another set called? $$A \subset\mathbb R$$ $$\frac{\sum A}{|B|}$$ I will try and be specific to my problem because I lack ...
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1answer
58 views

Similarity measurement for strings of letters

Let's say, I have 10 different groups, and each group has its own string sequence. So, it should be like : G1 -> CHFAIEBD G2 -> HCFJIGBD G3 -> HCFAIJBD G4 -> HFCIJEBD G5 -> ..... G6 -> .... The ...
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1answer
79 views

How to represent R function dexp mathematically?

I need help representing this line of code mathematically. exp(sum(dexp(x,prop_theta,log=TRUE)) - sum(dexp(x,cur_theta,log=TRUE))) Thanks for your help in ...
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1answer
17 views

Given MTTF and Number of Items, how to calculate failing parts with Time?

I am wondering how to make use of MTTF Here is the situation, I am given an MTTF for an item type x and a certain demand for that item in the next 25 years, say 100 parts that will be in operation ...
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2answers
130 views

Calculating the best match between two sets

I’m a PHP developer and I have a problem calculating the perfect match between two different data sets. I have data sets from companies, where each company defines the requirements for a specific ...
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56 views

Largest possible mean deviation

In a data set with 12 numbers, and the lowest and largest numbers can be 0 and 10, what are the numbers in the set to get the largest mean deviation. The mean deviation is obtained by: Finding the ...
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1answer
146 views

Likelihood ratio test rejection region for uniform parameters

Let $Y_1...Y_n$ be a sample from a uniform distribution over the interval $[0, \theta]$. For $H_0 = \theta_0$, $H_a = \theta_a$, $\theta_a \gt \theta_0$ we know that the most powerful $\alpha$-level ...
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1answer
48 views

Let X be a random variable with PDF fx. Find the PDF of the random variable |X| in the following

Here's my question: X is uniformly distributed in the interval $[-1,2]$. Find pdf of $|X|$... So I did P($|X| \le x$) = P($-x \le X \le x$)... From here I'm not too sure how to proceed. I know the ...