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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Consistent estimate using likelihood function

Let be $X$ a population with normal distribution. Using likelihood function I get the below expression $\hat{\sigma_X}^2 = \sum_{i=1}^{n}{\dfrac{(X_i-\mu)^2}{n}}$ for variance. I want prove that ...
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129 views

Is a variable transformed from a random variable a random variable?

Let a random variable $X$ be uniform distribution in $(0,1)$ and another variable $Y=X^2$. I calculate the sample space of $Y$ is also $(0,1)$, and the cdf and pdf of $Y$ are $F_Y(y)=\sqrt{y}$ and ...
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131 views

Measuring Similarity of Affine Transformations

I am currently working on a problem where a calibration Algorithm provides me with an Affine Transformation that transforms a 2D Image to it's assumed Position in a 3D Volume. To evaluate the accuracy ...
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6k views

Defective items probability question.

Hi I'm working with probability as part of an engineering course, and I'm struggling with the following tutorial question: Components of a certain type are shipped to a supplier in batches of ten. ...
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1answer
27 views

Is there a statistic model that selects attributes in each state?

Every state has its own sub-attributes, for example, state S{ATTR1, ATTR2, ATTR3}. We can train the model use data with states filled with all three attributes. When we want to predict, we just want ...
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99 views

Statistic: calculating error of order of two sequences of objects

I am trying to derive a meaningful statistic from a survey where I have asked the person taking the survey to put objects in a certain order. The order the person puts the objects is compared to a ...
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1answer
111 views

Help with $E(X)^2 = \sum r^2 P(X=r)$

I'm using some Anki flash cards to study statistics and have come across an equation I don't recognize: $$E(X)^2 = \sum r^2 P(X=r)$$ I found a similar equation in the last line of the proof of the ...
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1answer
31 views

Partitioning a sequence of numbers based on a percentage

I have a sequence $s$ of rational numbers where $|s|=n$. I am given a percentage $p$ to partition the sequence $s$ into $\frac{1}{p}$ parts, such that each part contains $p\%$ number of elements of ...
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126 views

Maximizing the expectation of a function with a constraint

I am interested in computing the maximum of the following function $E[ ((XB) - E(XB))^TR ] $ subject to the constraint $E(XB)=1$ where $X$ is a $n\times m$ random matrix, $B$ is a $m \times 1$ ...
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1answer
93 views

Shows the number of children per family for 50 families in a housing estate

Shows the number of children per family for 50 families in a housing estate $$\begin{array}{cc} \textrm{No. of Children} &\textrm{No. Of Families} \\ 0& 4\\ ...
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301 views

How to read a histogram?

I posted the following question over on StackOverflow, and I think I am having a hard time understanding exactly how a histogram works. I have the following set of numbers between 0 and 9: ...
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3answers
431 views

Eigenvalue of $X(X^TX)^{-1}X^T$

I just discovered today that $\mathbf{1}=(1,\ldots,1)$ is an eigenvector with eigenvalue $1$ of $X(X^TX)^{-1}X^T$ for certain $X \in \mathbb{R}^{m\times n}$ where $n \leq m$ and where the first column ...
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2k views

Matlab: least square method

Specify a linear function in terms of the least squares method approximates the set point table. Then calculate the sum of squares deviations of this linear function in given points. xi=[1.25 2.34 ...
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1answer
1k views

Statistics: Bertrand's Box Paradox [duplicate]

Possible Duplicate: Probability problem This is the Bertrand's Box Paradox I read about on Wikipedia: Assume there is three boxes: a box containing two gold coins, a box ...
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1answer
91 views

Proving that the sum of Good-Turing estimators is $1$

I want to know how to go about proving that the Good-Turing estimator has a total probability of $1$. I have seen this proof (page 2) but I found unclear the first step: $$\sum_j \theta[j] = \sum_r ...
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1answer
122 views

proof using (fixed point theorem)

I am seeking to solve for a Nash equilibrium in pure strategies $(d_2,d_2)$ involving two players, $1$ and $2$. Given that $h'(.)$ is s strictly decreasing and continuous function, $\Phi(d_1-d_2)$ ...
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2answers
1k views

How to calculate the expected frequency of a pattern?

I'm working on a problem to find the expected frequency of a pattern. Say there is a sequence of alphabets - A, B, C and D. The sequence is: ABDACDBADA. I want to find the expected frequency of a ...
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1answer
353 views

Estimating the covariance matrix with a set of vectors for the Mahalanobis distance

I am trying to figure out how to use the Mahalanobis distance still. I am having trouble figuring out how to produce my own covariance matrix. I guess the relevant link is ...
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1answer
2k views

Finding the sample size given the 95% confidence interval bounds, standard error, and sample mean?

I am not given the standard deviation in this problem. I understand that the following holds when the data is normally distributed: Upper 95% bound $= (\bar{X} + (SE) \cdot 1.96)$ I have all of ...
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5answers
13k views

Correlation matrix from Covariance matrix

This is for a project which I've been trying to find some information for Covariance matrix and correlation matrix. I understand that for a $n \times n$ matrix $A, AA^T$ will give me the covariance ...
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1answer
133 views

Best Fit Line/Linear least squares fits

In my Physics intro/data analysis lecture, theres some mention of "Linear least squares fits": Gradient $$m = \frac{1}{\Delta} (N \sum x_i y_i - \sum x_i \sum y_i)$$ $$\sigma_m = ...
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1answer
114 views

expected value properties

I'm reading an algorithm proof in which there is a passage I can't understand. Substantially we have 2 statistically independent random arrays $X$ and $Z$, and a third random array $Y$. It's known ...
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1answer
76 views

Ways to compare a set of values against a base value

I have a statistics doubt. Consider I have the following values. ...
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1answer
53 views

interpretation of the conditional mean

If i have 2 random vectors $X$ and $Y$, each of them with own different pdf, what is the conceptual interpretation of saying that the conditional expectation of $X$ given $Y$ is null? i.e. ...
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1answer
76 views

How to calculate variance on spending?

My maths is a little rusty, but I understand how to calculate variance in general. A colleague asked me if I knew how to calculate the percentage variance on budget spending. Given this example: ...
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1answer
338 views

Proof of Bayesian Prediction Formula

I was wondering how to proof this formula, commonly used in Bayesian Prediction: $$ \mathrm{P}(x|\alpha) = \int_\theta \mathrm{P}(x|\theta)\mathrm{P}(\theta|\alpha) \, \mathrm{d}\theta$$ The left ...
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148 views

Using a weighted table?

I'm writing a simple computer program that has to use the following weighted table to pick a random "payment" where the relative probabilities are given by ...
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730 views

Fast variance calculation

Suppose to have a sequence $X$ of $m$ samples and for each $i^{th}$ sample you want to calculate a local mean $\mu_{X}(i)$ and a local variance $\sigma^2_{X}(i)$ estimation over $n \ll m$ samples of ...
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1answer
78 views

Looking for the Name of this property: $\mathsf{P}\left(X \leqslant x\right) = \mathsf{P}\left(h(X) \leqslant h(x)\right)$

$h(\cdot)$ denotes a strict monotonic increasing transformation such as $\log$. Another inequality I do not quite get is that $$\mathsf{P}\left(h(X) \le h(x)\right) \ge \mathsf{P}\left(X \le ...
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3answers
6k views

How to calculate the Mahalanobis distance

I am really stuck on calculating the Mahalanobis distance. I have two vectors, and I want to find the Mahalanobis distance between them. Wikipedia gives me the formula of $$ d\left(\vec{x}, ...
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1answer
201 views

Is there name for arithmetic mean divided by RMS average?

Is there a mathematical term for the ratio of arithmetic mean to the root mean square average? (FWIW, in the context where I'm concerned about this the component values will always be >= 0.)
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729 views

Finding \alpha and \beta of Beta-binomial model via method of moments

I am looking for a laymen step by step of how the process of finding the 1st and 2nd sample moments located: http://en.wikipedia.org/wiki/Beta-binomial_distribution#Maximum_likelihood_estimation ...
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681 views

Statistical Significance Dice Probability

I think I just need to be pushed to the right formula or algorithm... Imagine you've got a "real" dice which is not an optimal one. And you want to know the confidence intervals for each result. So ...
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339 views

What is the rationale behind the evaluation of the Expectation operator?

We know that the expectation operator is defined for a random variable $x$, as such: $$ \mathbb{E} \left\{x\right\} = \int_{-\infty}^{\infty} x \: p_x(x) \; \mathrm{d}x $$ Where $p_x{x}$ is the PDF ...
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2answers
68 views

Showing probable causality

After examining various correlations between longitudinal data and illustrating high correlation between one or more variables, I realized that I could only show that the data was correlated but could ...
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1answer
113 views

non-intuitive results in random sampling

I play a card game (Magic The Gathering) that involves creating a deck of cards that can loosely be split into two categories: land and spells. You are permitted to play only one land per turn, and ...
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1answer
1k views

Covariance matrix of a complex random variable

If one considers a complex random variable as the joint random variable of the real and complex part, the covariance matrix of two complex random variables $Z_{1}=X_{1}+iY_{1}$ and ...
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1answer
238 views

Approximation using Legendre polynomials

my aim is to fit data points by the use of Legendre Polynomials. Has anybody experience with this task? My final aim is to do this automatically with mathematica. Thanks, rainer
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1answer
164 views

Relationship between Vapnik-Chervonenkis classes of sets and functions

I'm reading van der Vaart's "Asymptotic statistics" and on page 275 the author introduces the concepts of Vapnik-Červonenkis (VC) classes of sets and functions and states a proposition which I can't ...
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121 views

Mixture of two mirror-image Gaussians

Suppose we are given a set of points $(x_i, y_i)\in\mathbb{R}^2$ and are told that they are drawn from a normal (Gaussian) distribution. It is a simple matter in that case to find the mean ...
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1answer
144 views

Notation question on empirical p-value computation

In calculating an empirical p-value for test statistic $T_{0}$ (in my case the KS test statistic, $D$) using a permutation test $\tilde{p}_{N}(x)=\frac{N\tilde{G}_N(x)+1}{N+1}$ $\quad$ (3.4) where ...
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5answers
360 views

Rating system incorporating experience

I am looking to rank players on something and right now I am using a win-loss percentage to "rank" the players. The issue is that I have some people who just beat out the 5 game threshold and have a ...
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2answers
900 views

Rounding of complementary percentages

This problem arose while tallying votes in a poll and displaying the results as a whole number percentage. So if "Option A" received $300$ votes and "Option B" received $100$, the display should read ...
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2answers
326 views

Two Sample Confidence Interval for Normal Distributions

Let's say I have two independent random samples $X_1, X_2, \dots, X_n$ and $Y_1, Y_2, \dots, Y_n$ from normal distributions with real, unknown means $\mu_x$ and $\mu_y$ and known standard deviations ...
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1answer
95 views

By subjects t test and wilcoxon signed rank test

I am trying to see if people pronounce one kind of word (say nouns like "cat") with a longer duration than another kind (say plural nouns like "cats"). I had $18$ people pronounce words from lists of ...
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1answer
97 views

Need to calculate significance from a data set.

I have a set of data that looks like this: ...
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1answer
250 views

Html Tag counting - Rate of Change formula

I've been trying to a find a statistics-esque formula for calculating the rate of change for html tags which are either added or removed from various websites. So, for example, with the scraper I'm ...
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71 views

Basic arithmetic question

I have to find $80\%$, $60\%$, $40\%$, $20\%$ and it's values of a range of scores. For one range, my values go from $2$ to $8$ (mean will be $5$ which means, $50\%$ of that range's value = $5$) For ...
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1answer
167 views

Odds of 1 out of 7 NFL teams making the playoffs

I have a potential prop bet in the making where my opponent has to pick 7 NFL teams not to make the playoffs for this upcoming season. If all 7 of their teams miss the playoffs, they win, if at least ...
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1answer
46 views

Checking whether or not a variable has impact

For a statistics task I've been given a data set (regarding drugs prevention) and a few questions. One of the questions is to check whether or not the choice of treatment, treatment A or treatment ...