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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Average distance between two randomly chosen points in unit square (without calculus)

Imagine that you choose two random points within a 1 by 1 square. What is the average distance between those two points? Using a random number generator, I'm getting a value of ~0.521402... can ...
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1answer
41 views

How to prove the sign test

Please correct me if I'm wrong, but a version of the sign test assumes under $H_0$ that there is some distribution $F$ where $X_i \sim F, Y_i \sim F$ and $X_i, Y_i$ are iid. Then it states that $T = ...
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3answers
36 views

Probability of a Rare Event Occurring within a Certain Number of Times

I'd like to know how to find the probability of an event occurring, given the probability of that event, within a certain number of chances for it to occur. For example, say that once every year ...
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1answer
80 views

How to minimize the expectation?

Given random variables $X_0, X_1, \ldots, X_n$ with finite expectations $m_0, m_1, \ldots, m_n$ I want to prove that the numbers $a_i = \frac{\det \Lambda_{i0}}{{\det \Lambda_{00}}}$ minimise the ...
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1answer
21 views

statistics testing a claim please help. homework problem and stuck [on hold]

Systolic blood pressure levels for men have a variance of 388.09 mm^2 HG. a random sample of 31 women resulted in blood pressure with a variance of 497.29 mm^2Hg. Use a 0.05 significance level to test ...
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15 views

Confidence interval from covariance matrix

We have a matrix of stochastic variables $X\sim\mathcal{N}(0,\Sigma^2)$, where $\Sigma^2$ is a positive definite covariance matrix. How do we calculate the 95% confidence interval for X? (lets say ...
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1answer
14 views

Bivariate Normal Distribution Problem vs Marginals

If $X_1 \sim N(0,1)$ and $X_2 \sim N(X_1^2, 1)$ then does $(X_1, X_2)$ follow a bivariate normal distribution? My thinking is that $X_1 ^2$ is $\chi^2_1 $ since it's the square of a $N(0,1)$ random ...
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27 views

Combination of historical and current data in statistics

I have a general question about a statistical matter. Lets assume there exists a true and unique probability $p$ such that an event $X$ happens in the next 12 months. There is some information about ...
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33 views

Algebra Integral simplification

Let some equation problem final result is like this $0\leq1\leq s\leq t\leq u\leq v$ \begin{align} M=\mathrm{exp}\bigg\{-\pi\lambda v^2+\pi\lambda v^2\bigg(\displaystyle\int_o^s ...
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1answer
30 views

Standard deviation: calculating how polarizing a question is

I'm trying to calculate how polarizing a question is. Let's say I have a question that has 3 possible choices. A certain percentage of people choose a specific answer. Answer a: $30\%$ Answer b: ...
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3answers
38 views

2 restaurants located randomly

any help on following question will be much appreciated. Q. Suppose that $2$ restaurants are going to be located at a street that is $10$ km long. The location of each restaurant is chosen randomly. ...
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1answer
98 views

Sum of i.i.d. random variables and finding an upper bound

Problem: Suppose that $(X_i)_{i\in\mathbb{N}^+}$ is a sequence of i.i.d. random variables. For some $n\in\mathbb{N}^+$, let $S_n=\sum_{i=1}^n X_i$. Furthermore, let $a$ be a positive constant, and ...
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1answer
28 views

Challenging probability and statistics problems?

The books I have on the subject lack entertaining problems. Can someone suggest a book with more challenging problems - perhaps not at olympiad level, but a bit easier. Thanks very much!
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26 views

Explanation of Cramer-Wold theorem

I was trying to understand mathematically what the statement of Cramer-Wold theorem means. Intuitively, I was told that two probability distribution $P,Q \in \mathbb{R}^n$ are equivalent if all their ...
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15 views

If the null hypothesis is true, how will the test statistic be distributed?

I went with T~(50-6) The question goes.... "A regression is estimated with 50 observations, five explanatory variables and with a constant. Suppose You want to test the following hypothesis $H_0: ...
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0answers
9 views

Confidence interval for mean using t-distribution (unknown pop.variance). Is the following statement true or false?

This is from an A-Level statistics paper. We are the given the lengths of 10 components and asked to calculate a 95% confidence interval for the mean. We are given that the lengths are normally ...
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3answers
45 views

Parameters estimation for gaussian function with offset

I've read the paper Least square fitting of a Gaussian function to a histogram by Leo Zhou on how to perform a Least Square Fitting of a gaussian function to a histogram. The Gaussian function used ...
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1answer
9 views

Topics under Model Based Cluster Analysis

Can anyone recommended topic(s) I could use for my thesis under "Model Based Cluster Analysis"? I initially used "Inference in Model Based Cluster Analysis" as my working topic but appears to be ...
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1answer
42 views

A CD players plays songs until a certain artist is selected.

Suppose you have a CD player with 10 CDs where 9 of those CDs belong to an artist A and 1 CD belongs to an artist B. Suppose this CD player will pick a CD at random, and from that CD it will pick ...
2
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1answer
29 views

Chocolatier sampler boxes problem: applying goal programming and mixed-integer programing to optimally compromise goals.

QUESTION: A boutique chocolatier is planning to make a number of sampler boxes, each containing $36$ chocolates. (Therefore the total number of chocolates should be divisible by $36$.) The ...
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1answer
33 views

Testing randomness

I'm looking for informations about randomness and especially - random numbers. I found some about random number generators, but for now, the question, that concerns me is how statistically differ ...
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1answer
11 views

Maximization of a statistical property of a subset of random numbers

I have encountered a maximization problem which could be formulated as a discrete mathematics problem arising from statistics, but I don't know where to start or which techniques could be applied to ...
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1answer
27 views

Finding yearly weather statistics from tomorrow's weather probability

I'm trying to solve this problem from a book, but so far I haven't found how to approach it... I made a graph, and tried to calculate some probabilities.. but nothing What should I do? Thanks!
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2answers
17 views

What is a decision threshold and how does it apply to a statistical power?

I'm pretty confused on what is actually going on in this section with hypothesis testing. As another note, the values below are computed using R. I have a homework problem that says: From the ...
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1answer
28 views

Why is the Venn Diagram Considered a Special Case of the Formula for Unions

$P( A \cup B) = P(A) + P(B) - P(A \cap B)$ Or in Boolean terms: $P(A + B|C) = P(A|C) + P(B|C) - P(AB|C)$ I've read a lot of information but still can't piece together why this would be considered a ...
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1answer
29 views

Is the product of $n$ Normal distributions also a normal distribution? [closed]

Is the product of $n$ Normal distributions also a normal distribution? I need not derivation just but answer.
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0answers
9 views

Consistent estimators/convergene in probability and slutsky

Let $m_n$ be a consistent estimator of $g(\vec\alpha)$ where $\vec\alpha = (\alpha_1,\cdots,\alpha_k)\in \mathbb{R}^k$ and $v_n$ be a consistent estimator of $f(\alpha_1,g(\vec\alpha))$. Suppose that ...
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1answer
15 views

calculating variance problem

When calculate variance, It's 1/2E((x-0.1)^2)+1/2E((x+0.1)^2)-E(x)^2 Is it right? So, the answer is 0.01 Am I missing something? And whenever I calculate the vatiance of the second estimator ...
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1answer
15 views

Statistics for correlations with many (0,0) values

Suppose you have a large but finite collection of tweets. You want to know whether talking about football tends to correlate with talking about basketball. You can generate a table for a few hundred ...
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1answer
25 views

The difference between a matrix valued random variable and an $n \times p$ matrix of data

So I am totally new to the field of random matrices, but I was not sure about how they are applied. According to Wikipedia, a random matrix is "a matrix-valued random variable—that is, a matrix some ...
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31 views

bound on expectation of a two-variable function under an independent distribution

Consider a probability distribution $P(x)$, a set observed samples $S = \{x_1,\cdots, x_n\}$ where $x_i \sim P(x)$ for $i \leq n$, and a symmetric function $h(x,y)$. How can one efficiently ...
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43 views

Changing results of a random process

A very sexist population prefers boys to girls. All parents try various strategies (but not neglect, abuse, or selective abortion) to increase the number of boys, such as giving birth repeatedly ...
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1answer
20 views

variance and sample confused

when solving (b) Is variance $$V(\frac{1}{2}(x_1+x_2)) = \frac{1}{4}V(x_1+x_2)= \frac{1}{4}(v(x_1)+v(x_2))= \frac{1}{2}\sigma^2$$ or should I divide variance by the sample size so that ...
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1answer
33 views

Trying to understand a Mean and Standard Deviation Question

I am trying to understand this simple question relating to mean and standard deviation. "I currently have $100$ employees in my firm and have found that the growth rate in the next $5$ years will be ...
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1answer
24 views

statistics - estimator and biased unbiased [closed]

I am having a problem with this my solving proceducre is that $E(\theta)= 1/2E(X-0.1)+ 1/2E(X+0.1) = 1/2$ So, $E(\theta)1/2 - (\theta)1/2 = 0$ which means it is unbiased. Variance is ...
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1answer
20 views

How to compare dispersion of data?

From my statistic book, I learn that standard deviation is used to estimate how much the data spread around the mean value. If I have two or more sets of data, for each one, they normalized to the ...
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1answer
11 views

Linear or Non-Linear Model

I have the following regression equation \begin{align*} y_i = \alpha + \gamma\cdot\beta\cdot x_i+ \varepsilon_i, \end{align*} where $y_i$, $x_i$ and $\varepsilon_i$ are $n\times 1$ vectors, ...
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0answers
18 views

R Studio help - qqnorm [closed]

I'm loading in a file called data and using the command qqnorm like this data <- read.table("data.txt", header=TRUE) qqnorm(data) And get the error message "Error in xy.coords(x, y, xlabel, ...
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1answer
10 views

Calculating cumulant from data set

Given a data set $x = \{ x(n) | n=1,\cdots,m\}$ how do I calculate the $p^{th}$-order cumulant? In particular I need to calculate the 4th-order cumulant. I found that I can calculate the cumulant from ...
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1answer
17 views

Statistics: Expected return of investment

I want to solve the following exercise: We have a budget of 10 000 [some currency]. We divide it into two parts, w and 1-w. The part w has an expected return of 8% with a standard deviation of 25%, ...
2
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1answer
23 views

Direction of Study Suggestion

I'm not sure if this belongs in stats or here, or why stats would be considered different to math. During my self-study of biochemistry and medicine, I notice that a vast range of published studies ...
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1answer
51 views

Infinite population mean?

When reading about the central limit theorem, the concept of infinite population mean arises.How can a population mean be infinite?
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1answer
28 views

Upperbound a logarithmic expression that has a covariance matrix

Let $\Sigma$ be a $2\times 2$ covariance matrix and ${\bf h}$ a vector of complex values entries. $$A= \log(1+ {\bf h}^* \Sigma {\bf h} )$$ $$\Sigma = \begin{bmatrix} 1-|\rho_1|^2 & \rho_3 - ...
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15 views

Point Estimation.

11 Lecture about Point Estimations 11.1 The Tasks a) We measure the pocket money of visitors at the game fair in Leipzig at three days. Estimate the means, estimate the overall mean and compare it ...
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1answer
23 views

Poisson Distribution Problem

The random variable $X$ has a Poisson distribution with mean $\mu$. Show that $$P(X \equiv 1 \bmod 2)=a+be^{c\mu}$$ where $a$, $b$, and $c$ have to be determined. I plugged $X \equiv 1 \bmod 2$ ...
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3answers
26 views

Test statistics, critical value

"What is the critical value of the test statistic?" Does this question even make sense? From my knowledge test statistics is one thing and critical value is another thing. If test statistic is larger ...
2
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1answer
28 views

Conditional expectation and rao-blacwell

I am studying on UMVUE, and I'm struggling to find that conditional expectation Let $X_1,\ldots,X_n$ random sample of $X\sim U[0,\theta]$. i) Show that $2X_1$ is a unbiased estimator for $\theta$ and ...
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0answers
7 views

Ellipsoid confedence intervales?

Are Bonforonni, Scheffe, Multivariate t, and Tukey for simultaneous Confidence intervals are ellipsoid? How can I tell from the form of the interval that it is ellipsoid or rectangular?
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1answer
14 views

Correctness of a set with respect to another set.

Is there a specific measure for correctness of a Set w.r.t another set? e.g. Consider there's a base set A, and a set B whose correctness needs to be measured w.r.t set A. Now B might contain some ...
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0answers
21 views

Rao-Blackwell theorem and uniform distribution [on hold]

Let $X_1,...,X_n$ random sample of $X$~$U[0,\theta]$.Use the fact that $X_{(n)}=max(X_1,..,X_n)$ is a sufficient and complete statistic and Rao-Blackwell theorem for show that ...