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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Distribute Cakes Problem

Describe: In one party, there are N cakes, the volume is V1, V2, V3,..., Vn, each cake has an unique flavor. There are K people in this party, we want to divide cakes and distribute them to these ...
2
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28 views

Does an exponential model fit my data?

I am measuring accumulation of a fluorescent-tagged protein at a particular location within a cell over time. In previous experiments that I have performed, I see a standard exponential distribution ...
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1answer
10 views

time series similarity heuristic

What is a good heuristic for time series similarity. I'm ok with false positives bc I will then pass the results of the heuristic to an actual series similarity algo. Basically I want the heuristic to ...
2
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1answer
44 views

Math required for medical statistics

I have never been good with Math. I talk to many of my medical colleagues and it seems to me that most of them have a poor understanding of statistics. Many of them claim to understand it but actually ...
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1answer
16 views

How to calculate lower & upper quartiles?

I'm sure this has been asked many times before but it's confusing me a lot so hopefully someone can help! I am given this data set: 0.28 0.30 0.42 0.59 0.71 1.67 1.82 2.39 4.71 4.79 4.89 5.00 5.00 ...
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2answers
60 views

How to give rigorous proofs of these two limit statements?

Let $X$ be a random variable with cumulative distribution function $F(x)$. Then how to rigorously prove the following two limit statements? $\lim_{x \to - \infty} F(x) = 0$. $\lim_{x \to + \infty} ...
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1answer
19 views

Why the standard deviation approximated by the range rule of thumb much different from the standard deviation calculated directly from sample data?

What makes the standard deviation approximated by the range rule of thumb much different from the standard deviation calculated directly from sample data? For example, if the approximate standard ...
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24 views

Outliers in a Normal Distribution

Im doing AP Stat. in High School level. Here is a question i am stumped on because i feel like it is maybe a threory or law or something that i just never learned. However it DOES ask to show my ...
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14 views

Finding moment estimator

I am required for a homework question to find moment and maximum likelihood estimators, and in class we have only looked at finding MLEs and using already known moment estimators, hence I have no idea ...
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1answer
19 views

Find the Distribution of an Order Statistic

I am working on a practice prelim question. It states: Let $Y_1 < Y_2 < Y_3 < Y_4$ denote the order statistics of a random sample of size 4 from a distribution which is uniform on ...
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1answer
23 views

Interpretation of a statistical formula involving the ratios of sample and population

Kind of an odd question, but, is this a standard equation for stats? I can't figure out for the life of me what it looks like. $\left(\frac{\text{Sample Of A}}{\text{Population Of ...
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1answer
15 views

Calculating expectation and using the delta method

Let $\hat{p}$ be the proportion of successes in $n$ independent Bernoulli trials each having probability $p$ of success. (a) Compute the expectation of $\hat{p} (1-\hat{p})$. Isn't $P(\hat{p} = k) = ...
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1answer
9 views

Derivation of Mode of grouped data

A formula to calculate the mode for grouped data's is given in my text book: Mode = $l +$ $ (f_1 - f_0) \over (2f_1 - f_0 - f_2) $ $ * \ h $ Where, $l = $ lower limit of the modal class, ...
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1answer
33 views

A Bayesian estimate of the bias of a coin

Consider a coin with unknown probability $p$ of landing on head. I will toss the coin and stop as soon as I get a head. Say this is after $n$ tosses. If my prior belief for $p$ was uniform on ...
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1answer
22 views

$|p- \dfrac xn|>|q- \dfrac xn|$ $\implies$ $p^x(1-p)^{n-x}<q^x(1-q)^{n-x}$?

If $p,q \in (0,1)$ , and $ n \in \mathbb N$ be given and $x$ be given integer between $0$ and $n$ such that $|p- \dfrac xn|>|q- \dfrac xn|$ , then is it true that ...
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1answer
26 views

Estimator with smallest variance, Lagrange multiplier

I have a question and I think I'm supposed to use the Lagrange multiplier although I haven't been taught it, so I'm not sure if I can use it or not. The question is: Suppose that $X_i$ has mean $\mu$ ...
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0answers
5 views

Showing deviation of data on a fixed scale

I'm creating a chart in D3 which shows the Mode from a large (1mil +) dataset; and colors it according to the standard deviation of the dataset. But the issue is that I need to map the range (Red ...
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1answer
44 views

Maximum likelihood estimators

I have $X_1,X_2,\dots,X_n$ as random samples from a binomial distribution, with probability function: $$p_X(x) = Pr(X=x) = {m \choose{n}}\alpha^x(1-\alpha)^{m-x},x=0,1,2,\dots,m$$ where $m$ is given ...
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13 views

Expectation Maximization Algorithm for Gaussian Mixture Model

Can we use the Expectation Maximization algorithm for estimation of Gaussian Mixture Model with full covariance matrices? If yes then can you please give me a reference paper? So far all the machine ...
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4answers
73 views

Estimate bias of a coin

Consider a coin with probability $p$ of landing on head. You can estimate the prob by tossing it lots of times and looking at the proportion of heads one gets. In my problem I just want to know if ...
0
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1answer
25 views

Is Inverse Standard Normal Distribution $\Phi^{-1}(x)$ related to $erf^{-1}(x)$?

I am looking to use $normsinv$ to implement $inverf$ in a spreadsheet. Cumulative Standard Normal Distribution, $\Phi$, is related to $\mathrm{erf}$: $\Phi(x) = {1 \over 2}[1+\mathrm{erf}({x \over ...
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10 views

An example of $k$-independent distributions.

I'm trying to better understand the idea of $k$-independence in distributions. The idea is that a distribution $\mu$ over $\{0,1\}^n$ is $k$-independent if any restriction of $\mu$ to $k$ variables ...
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0answers
24 views

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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18 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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6 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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12 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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23 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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21 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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22 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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0answers
16 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. ...
2
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1answer
31 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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23 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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1answer
16 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
14 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 ...
4
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1answer
49 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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1answer
64 views

stats probability [on hold]

the typical Australian tropical cyclone season 1 November to 30 April averages around 12 tropical cyclones. we assume the prob of arrival of a cyclone is the same for any given month of season a ) ...
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1answer
25 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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0answers
25 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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26 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
37 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 ...
3
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1answer
49 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 ...
2
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0answers
40 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 ...
2
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1answer
17 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
34 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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0answers
13 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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12 views

Conditional mean of gumbel random variable [on hold]

Let $x_1$ and $x_2$ be independently drawn from standard Gumbel distributions. How can I show that $E[x_1 | x_1>x_2+\mu] = \gamma-\log\Pr(x_1>x_2+\mu)$, where $\gamma$ is Euler's constant? ...
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1answer
16 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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18 views

Finding the sample size

My main question is this: Assume I want to conduct a plebiscite of an entire town about whether or not the citizens want to have their water fluoridated or not, I want to conduct a survey first. If ...
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10 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
22 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: ...