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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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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Algebra Integral simplification

Let some equation problem final result is like this \begin{align} M=\mathrm{exp}\bigg\{-\pi\lambda v^2+\pi\lambda v^2\bigg(\displaystyle\int_o^s \frac{2x}{v^2}\mathrm{d}x \nonumber \\\\ ...
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1answer
34 views

Coin Flips and Hypothesis Tests

Here's a problem I thought of that I don't know how to approach: You have a fair coin that you keep on flipping. After every flip, you perform a hypothesis test based on all coin flips thus far, with ...
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13 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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2answers
24 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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3answers
27 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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18 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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18 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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13 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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21 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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3 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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1answer
289 views

Finding the regression line given the mean, correlation and std dev of x and y.

So we 100 observations for (x, y). The mean of x is 1.06, and for y it is 3. The std dev is 0.52 for x and for y it is 1.13. the correlation between x and y is 0.89 In the question we are told to: ...
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1answer
45 views

Matrices - Inverse of the principal square root of a covariance matrix (^-1/2)

Say you have a square (variance)covariance matrix S How would one go about working S^-1/2 (inverse of the principle square)? Bearing in mind, I'm trying to understand a paper which states: I've ...
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1answer
16 views

goal programming, mixed-integer program, optimal compromise goals, statistics

QUESTION Can someone help me figuring out how to calculate this question? In this question, I have 4 variables (I think it would be more easily to calculate), and 5 goals. SO for the 27 boxes, I ...
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2answers
38 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
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 ...
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1answer
7 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
28 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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21 views

Maximizes the expected utility, Decisition making, statistics

QUESTION Can someone help me figuring out how to calculate this question, i just learned this stuff and i haven't do any example like this before... So i'm interpreted this question as: We need ...
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1answer
326 views

Confidence interval of quotient of two random variables

I have random variables $X_1, X_2, \dots, X_n$ and $Y_1, Y_2, \dots, Y_n$, with $n$ a large integer. All pairs $(X_i, Y_i)$ are independent and identically distributed, but every $X_i$ and $Y_i$ ...
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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
216 views

Conditions for positive dependence

Consider two random variables $X$ and $Y$ with joint distribution $F_{X,Y}$ and strictly positive density function $f_{X,Y}$. Additionally, let $x^*$ be the value of $x$ that solves: $$ \Pr[Y\leq ...
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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
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
26 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
19 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
22 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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1answer
29 views

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

Is the product of $n$ Normal distributions also a normal distribution? I need not derivation just but answer.
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29 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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1answer
881 views

How are the values in the Shapiro Wilk weight table calculated?

http://www.real-statistics.com/statistics-tables/shapiro-wilk-table/ I am trying to build a function that performs a Shapiro-Wilk test on a sample population, but I cannot figure out how the a values ...
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1answer
21 views

statistics - estimator and biased unbiased [on hold]

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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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
386 views

UMP of a Beta($\theta,1$) distribution

I need to find the UMP of a BETA$(\theta,1)$ distribution. I know that the pdf of this problem is $$f(x;\theta)=\theta x^{\theta-1}=\theta e^{(\theta-1)\log{x}}$$ Thereby this family contains a MLR ...
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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
14 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
49 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
29 views
+50

How do correlation and causality effect this scenario?

I came up with a real world problem that I don't need to solve, but was intrigued by nonetheless. Imagine I was trying to figure out if a supermarket chain took credit as well as cash. My wife had ...
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1answer
32 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
25 views

Unequal measurements

Let $\mathbf W \in \mathrm{Mat}_{n\times m}([0, 1])$ be a full-rank matrix ($n\geq m$). Suppose the following equality holds: $$\mathbf y = \mathbf W \mathbf x,$$ where $\mathbf x, \mathbf y$ ...
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1answer
18 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
8 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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2answers
37 views

Piece-wise probability density and cumulative distribution function exercise

Given a random variable $X$ with the density function: $f(x) = a$ if $0 \leq x \leq b$ and $f(x) = b$ if $b < x < a + b$ I want to solve the following exercises regarding this ...
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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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18 views

R Studio help - qqnorm [on hold]

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
50 views

Conditional probability to Conditional expectation

My first related question is this link. enter link description here Expected value of geometric distribution is $$c=E(X)$$ $$c=0\times p+(1+c)\times q =q+cq$$ $$c=\frac{q}{p}$$ when $X= number \ of ...
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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%, ...
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1answer
24 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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1answer
9 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
29 views

Confidence Interval for Nonlinear Regression using F-Test - lmfit

I am trying to understand the implementation for the lmfit confidence interval calculation - in the docs it is stated: "The F-test is used to compare our null model, which is the best fit we have ...