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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measure of dependence for copula

I have some question about the paper of Schweizer and Wolff (1981). The question concerns about the following bound $$\int_0^1\int_0^1|C(u,v)-uv|\,du\,dv\leq\frac{1}{12}$$ where $C$ is any copula. ...
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The definition of the sample standard deviation

I am reviewing the statistic. I read the book "Probability and Statistical Inference" which is written by Robert V. Hogg and Ellit A. Tanis. There is a statement in the book says that: (Section: ...
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Features of a Set Given Same Value for Maximum and Average

What else, if anything, can be said if the maximum of a set is equal to the average value in the set? If the average is equal to both the maximum and minimum then all the values are the same, but can ...
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Stats “AND” Elimination

What's the logical behind eliminating this AND in a solution? p(he lives to 80 GIVEN he lives to 50) = p(he lives to 80 AND he lives to 50)/p(he lives to 50) = p(he lives to 80)/p(he lives to 50) ...
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Definition of Time Series

Having not done any stats for a few years, I seek clarification regarding the definition of time series given in my textbook. I apologize for the length, but I would be glad to just resolve my main ...
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What can we say about correlation coefficients?

If we are looking at sales and inventory data with a correlation of 50%, what can I conclude? My intuition tells me that 50 % of the movement in one of the variables is attributed to the others ...
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generating random samples with a PDF

I have the PDF of a distribution from which it is not possible to get a closed from for the CDF or inverse CDF. Is there a technique that would allow me to generate samples from this distribution ...
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Correlation Statistic with a neutral score

I'm trying to develop a correlation statistic to compare lines/random variables (represented as a series of points). I want the correlation value to convey little information if one the lines is a ...
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Statistics. Chapter 8

A random sample of 50 items result in a sample mean of 20. The population standard deviation is 10. What is the standard error of mean. At 96% confidence, what is the margin of error. Compute the 98% ...
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Problematic Permutation Problem

i see a problem without any definition. would you please help me? i want to calculate the number of permutations of 1,2,...,1392 that 696 numbers be in the natural positions (from all numbers, 696 ...
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33 views

How to simulate from a simple point process

Define a point process by the conditional intensity function $$\lambda^*(t) = \mu + \alpha \sum_{t_i < t} e^{-(t-t_i)}$$ where $\mu$ and $\alpha$ are positive parameters. I would like to ...
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Statistically determine whether a set of variables are diverging

I'm collecting data on a battery which is being discharged and charged again repeatedly. This battery consists of 4 cells in series and I am recording the fully-charged voltage of each cell at the end ...
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Find the mean and median help!

I feel like there isn't much information, but the answer for the mean is 79. The median is said to not be able to be found. Can someone explain this? Thank you!
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“Approximation” of a maximum likelihood confidence set

I have some trouble constructing the "approximate" set, how can it be defined or calculated? Suppose that three characteristics in a large population can be observed according to the following ...
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Difference between Kendall tau distance and Kendall tau rank correlation coefficient

What is the difference between Kendall tau distance and Kendall tau rank correlation coefficient? Which one I should use to calculate how similar two arbitrary permutations of ranks are? Suppose, we ...
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27 views

Alternatives to absolute error?

Let me explain my scenario in which I need to calculate absolute error. Lets say the X is the actual value. And X' is the value of X with some error 'e'. So X' = X + e'. Lets say i = 1 to 10000. I ...
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Closed form equation for win percentage of two battling armies

I was pondering a battle mechanic for a board game that is similar to, but simpler than battling armies in Risk. Consider one army of size X and a second army of size Y. The battle occurs by ...
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51 views

Probability Question involving Probability Mass Function/Random Variables

Problem: When a paging system transmits a message, the probability that it will be received correctly by the appropriate pager is p. To ensure that the message is correctly received at least once, the ...
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31 views

Comparing the width or variance of two distributions with uncertainty on variance estimate

So I have two probability distributions ,one from data and one from a model. They are not Gaussian shaped but skewed. I have measured their rms or width by finding the median and the 16% (lower limit) ...
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36 views

Derivative of an exponentially weighted moving average

It has been a while since my university math courses, so let me apologize right off the bat... I'm using GSL to perform non-linear regression analysis and am mostly happy with the outcome, however, ...
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15 views

Difference between ANOVA and Pearson Chi-squared test

To test the equal of variance the non categorical variable using ANOVA test. As for the data with categorical variables using Pearson's Chi-squared. In this case, why the ANOVA test can not be used ...
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Will statistical analysis of transformed data hold for the original one?

I have a data with distribution like chisq-squared one. But ANOVA and t-test need the data to be normal distributed. So I want to do the Box-cox transformation to the data, but my concern is will the ...
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18 views

A simple plot of a list in R

This is a very easy question, I guess. But unfortunately I struggle to solve it in an easy way: I have a list in R. I would like to plot the entries of each column vertically and label the x-axis to ...
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19 views

UMVUE using complete and sufficient statistic

Let $X_1,X_2,...,X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. I showed that $(\bar X,S^2)$ is jointly sufficient for estimating ($\mu$,$\sigma^2$) where ...
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Maximum Likelihood estimators in linear models

Consider two simple linear models. $y_{1j}=\alpha _1+\beta_{1}x_{1j}+\epsilon_{1j}$ and $y_{2j}=\alpha _2+\beta_{2}x_{2j}+\epsilon_{2j}$ , $ j=1,2,...,n>2$ where $ ...
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34 views

Find a 10% likelihood interval

The function is: (n choose x)$[(1-y)^{k}]^{x}[1-(1-y)^{k}]^{n-x}$ Suppose n = 100, k = 10, x = 89 I found the maximum likelihood of y-hat to be 0.0116 Now I need to find a 10% likelihood interval. ...
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Two Reflecting Barriers

A chain with stats 1,2,....,n has a matrix whose first and last rows are (q,p,0,...,0) and (0,...,0,q,p). In all other rows Pk,k+1 = p, Pk,k-1 = q. Find the stationary distribution. I am ...
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How could you find the probability that the estimator is within 0.03 of the mean?

p = fraction of large population that smokes n = sample size y = # in sample that smoke The maximum likelihood estimate of p is p-hat = y/n Consider the random variable Y and estimator F = Y/n ...
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Simple question about showing independence [closed]

How does one show that it is possible for a random variable $Z$ to be independent of $A$ but also not independent of $X$ where $X=1\{A>B\}?$ Under what circumstances can this be true, ignoring the ...
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Conditioning on information about the moments of a random variable is trivial

Say we have some random variable, $X$. Is it always trivial to condition on information about the moments of $X$? For example, suppose we know that $\mathbb{E}(X)$ is positive. But ...
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Density of k-th order statistic

I know that the k-th order statistic density for the uniform distribution is given by: $f_{(k)}(y) = \frac{N!}{(N-k)!(k-1)!} t^{k-1}(1-t)^{N-k}$. This is the density if the $N$ samples are sorted in ...
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34 views

Finding UMVUE for Poisson distribution using Rao Blackwell

Let ${X_1,X_2,... ,X_n}$ be a random sample from a Poisson distribution with parameter $\lambda$.Let $\gamma(\lambda)=P(X<=1)$. Find UMVUE for $\gamma(\lambda)$. This is my attempt: First I ...
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What is the minimum number of data points required to use ARIMA(p,d,q)? [migrated]

I'm using the $\mathrm{ARIMA}$$(p,d,q)$ model to predict future time series data in R (see also R's ARIMA documentation). Q: What is the minimum the number of data points we must have in the time ...
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How to get the standard deviation of a given histogram (image)

I have a doubt: How to get the standard deviation of a given histogram? I'm looking for it on the internet. But I got Nothing. For example the case of this image below Thanks in advance!
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Statistical methods of calculating weights for criterion describing quality.

In a research of quality of different sound recording techniques volunteers where asked to grade samples using multiple criteria. Those criteria describes different aspects that directly contribute to ...
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Generating random variates in Excel

I am very confused with a question I have found in relation to Excel. I am hoping someone can help me do this or at-least give me direction in which I can figure out how to do this. So far I don't ...
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Infimum of Gamma distribution

Let $X$ be a Gamma random variable with the CDF $F_X(x)=\frac{1}{\Gamma(\alpha)}\gamma(\alpha,\beta x)$ where $\Gamma(x)$ represent the gamma function and $\gamma(a,b)$ denotes the lower-incomplete ...
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Find probability for distribution of several random variable

let $f(x,y)=k$ if $x >0, y>0, x+y <3$ and $0$ otherwise. Find $k$, find $P(x+y \leq 1), P(Y>X)$ I know that $0 <y<3-x$ but I also need to find the upper bound of $x$ in order to do ...
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Covariance of random numbers not near zero

I'm learning about making an efficient bounding volume for a point cloud and this touches on statistics: an area of mathematics I'm very unfamiliar with. The book talks about how to compute ...
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Find probability in card question

Suppose that a test for extrasensory perception consists of naming (in any order) $3$ cards randomly drawn from a deck of $13$ cards. Find the probability that by chance alone, the person will ...
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Probability of poisson distribution

Let $X$ be the number of cars per minute passing a certain point on Sunday from 8AM to 10AM. Assume that $X$ has poisson distribution with mean $5$. Find the probability observing $4$ or fewer car in ...
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What is the expected value of max(x-y,0)?

For a normally distributed $X$ (with mean zero and variance $\sigma^2$) and a constant $Y$, I need to know $\mathbb{E}(\max(X-Y,0))$ I think that this should be $\mathbb{P}(X<Y)\times 0 + ...
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Estimating an unknown addition to data set

I have a data set with 11 points. The smallest point is 60, the largest is 122, and the average is 92.36. This is all that is known about the data set. I add one point (unknown size), such that there ...
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Rate of convergence of a martingale

I have a question related to convergence rates of martingales: Assume that there is a sequence of maximized likelihood ratios: $ \frac{f_{\hat{\theta}_{n}} \left ( Y_{1},Y_{2},\dots,Y_{n} \right ) ...
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Distribution Significance

I am trying to determine if a distribution is significant given an overall population distribution. The population as a hole has a 4% distribution for my question. If a sub-group of my population has ...
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Find probability

Let $X$ (millimeters) be the thickness of the washer. Assume that $X$ has density $$f(x)= \begin{cases} kx & \text{if $0.9 \leq x \leq 1.1$};\\ 0 & ...
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Draw and compare the likelihood using R

The following shows the heart rate (in beats/minute) of a person, measured throughout the day: 73, 75, 84, 76, 93, 79, 85, 80, 76, 78, 80. Assume the data are an iid sample from ...
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Generating a given length sample and skewness whose normality is verified by one normality test but not by an other

I just want to generate 1 sample of length$=n>30$, |skewness$=S|<0.3$ and for which normality is not rejected by Shapiro wilk test of normality but rejected by Anderson darling test of ...
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Gamma distribution Norming constant for extreme minima

the norming constants for extreme maxima of Gamma distribution is known and is give in link.springer.com/article/10.1007/s10687-010-0125-3. I would like to know is there reference or paper that states ...
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Mathematical Probability and Statistics( all the math need)

I would like some suggestions about mathematical techniques and knowledge are required to understand and master 2nd year undergraduate probability and statistics. I am mature student with some ...