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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Some expectations of psi (digamma) function

I want to derive an Expectation-Maximization algorithm for my model. But some expectations of psi (digamma) function is needed in the procedure. Assuming I have a Gamma distributed random Variable ...
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what is the influence of the specific statistical model selection in a practical project

I hope this is the right place to ask this question. But if it is not, please feel free to migrate. There is a famous quote, which is like "all models are wrong, but a few are useful". So, I was just ...
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64 views

Deal with circular standard deviation

I have a sensor that gives me a direction of the target in degrees with a certain standard deviation and I use a circular probability density function (PDF) to represent probable locations for the ...
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13 views

Coarse Graining: Correlation destroyed after coarse graining

X Y are two correlated random variable. How likely the correlation between them will be destroyed if the following coarse graining method is applied on them? We divide the range of X and that of Y ...
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21 views

The probability density of samples

I've found many ways to estimate the pdf of the distribution you take samples from. However I did not find any ways to calculate THE probability density of the samples itself. How can I do that? ...
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10 views

Best linear predictor of an AR model

I am how exactly should understand the best linear predictor for an AR(p) model. For instance if we have an AR(1) model defined as $X_t = \phi_1X_t+Z_t$ with $Z_t$ is WN($0$,$\sigma^2$) and I want ...
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16 views

How to distinguish different states?

The linear model for sparse coding is y = Ax, where x is a sparse vector (m*1), A is the dictionary matrix (n*m), and y is the observation vector (n*1). If ...
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17 views

Correlation between ordinal variables

I want to know the correlations between 3 groups of variables which are oridinal (rating of 1 to 10). I have followed the formula on wiki and computed the Kendall tau-b correlation coefficients ...
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22 views

At which value is the density in two poisson-processes the same?

Two poisson processes, $X$ with $\lambda=3$ events pr. hour and $Y$ with $\lambda=5$ events pr. hour. Let $Y$ be the waiting time. At what value, $y$, in minutes and seconds, is the density, ...
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Gap test for RNGs

Gap test is discussed here gap test for random numbers , if we have probabilities how we can find the expected number of gaps in order to compare with counted number of gaps after running random ...
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40 views

Statistical distribution methods

I'm coding a program to classify a stock price movement in a given timeframe. I have around 20k data points and it ranges from 0 to around 300, with the average being 10,8. I want to use a genetic ...
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44 views

Mean and standard derivation

A set of numbers consists of one's and three's. Find the mean and standard derivation if there are 23 one's and 17 three's. What's the meaning of this question ? I know the formula for mean and s.d. ...
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14 views

Calculating Margin Of Error

I'm having trouble calculating the margin of error for this problem. "What is the overall margin of error of the survey, at 95% confidence, if it is based on a simple random sample of 1111 voters?" ...
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30 views

Percentage and stats

In Biology class you have $4$ tests, each worth $15\%$ of your final grade, and a final exam worth $40\%$. If your test scores are $80$,$80$,$95$ and $65$; and your final exam score is $85$, what is ...
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16 views

Is the SemiCovariance matrix positive definite?

I am trying to proof whether the SemiCovariance matrix defined in equation (5) from the following paper http://web.iese.edu/jestrada/PDF/Research/Refereed/MSO.pdf is positive definite or not. As a ...
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25 views

hypothesis test type 1 and 2 error

Supporters claim that a new windmill can generate an average of at least 800 kilowatts of power per day. Daily power generation for the windmill is assumed to be normally distributed with a standard ...
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20 views

multivariate interval estimation

I have several samples of probabilistic vectors, i.e, each sample is of the form $(x_1, \cdots, x_n)$ such that $\sum_{i=1}^n x_i\leq 1$ (they are sub-probabilistic vectors), how can I obtain a ...
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28 views

p value and hypothesis testing

In a random sample of 99 games the home team won 57 games. Test the null hypothesis That home team wins one half of all games against the alternative that the home team wins a majority games. How can ...
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How to combine 2 different variables (time&cost) into a fitness function?

We are working on a simulation of uber taxi drivers, and to measure the effectiveness of additional functionality (like picking up extra passengers along the way) we would need to create a fitness ...
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Hypothesis test, p value problem self study

In a random sample ,p=57/99 Hypothesis test H0 : P = 1/2 H1 : P > 1/2 I want to test the H0 and want to find the p value and threshold of significance level. I am trying to solve this problem. I ...
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19 views

What statistical methods can I use to compare the differences between the two time series?

I conducted gas emission measurements from two buildings for a year (one is the building without treatment; the other is the one with treatment). They are two time series data. I plotted the data and ...
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42 views

Combining Sample Means and Variances

Let us assume we have several samples of unknown size but known mean value $\mu_{i}(x)$ and known variance $\sigma_{i}^{2}$. Now we want to calculate the mean value and variance for the total ...
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15 views

Does Weighted avg of Weighted avg makes sense..?

Scenario is like we have a city in which there are many schools. All of them conducted a mathematics exam and it depends on the school for each class how much marks the exam would be. Suppose school ...
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What is Bootstrapping in statistics? How can I use it to determine error in the mean, variance, kurtosis and skewness of a data set?

From what I understood from searching randomly is that it has something to do with resampling. What does this resampling mean? Is it selecting random data from a distribution or is it getting data ...
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13 views

Explaining the reasoning behind the initial derivation steps for the Gamma distribution

When deriving the gamma distribution, how is it justified that $P(W>w)$ is the summation of separate exponential distributions? I understand that for the gamma distribution, we have a series of ...
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Independence - Probability and Statistics

Any help on this problem is greatly appreciated! I'm completely stuck School board officials are debating whether to require all high school seniors to take a proficiency exam before graduating. A ...
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31 views

$\overline{X} \rightarrow c$ in probability $\Rightarrow g(\overline{X}) \rightarrow g(c)$ in probability

Prove that if $\overline{X} \rightarrow c$ in probability and if g is a continuous function, then $g(\overline{X}) \rightarrow g(c)$ in probability. Once I think in the situation $|g|<M$, it is ...
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43 views

finding min and max after removing percentage of num values knowing the standard deviation.

I have a question i have some data and i know it's (number of values, min, max, mean and standard deviation) can I know the minimum after removing x% of the total number of values and the maximum ...
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31 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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44 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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70 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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16 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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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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34 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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34 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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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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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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38 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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33 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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28 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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61 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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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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9 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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17 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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1answer
26 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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How to find the mode when none of the terms repeat?

What is the mode of a series / distribution of values when none of the values of the distribution are repeated , for example : What is the mode of 1 , 3 , 45, 67 , 89 , 100
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Bayes estimates

How do I attempt at solving this problem? Could I use proportionality? Bayes estimate of parameter of lambda with Poisson likelihood with x = (1,5,4,4) and gamma prior for lambda with mean = 2 and ...
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53 views

Proof for Mean of Geometric Distribution

I am studying the proof for the mean of the Geometric Distribution http://www.math.uah.edu/stat/bernoulli/Geometric.html (The first arrow on Point No. 8 on the first page). It seems to be an arithco ...
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9 views

Can an average calculated with a neutral weight be different from a simple average?

I'm working with a data set and just found that some averages calculated with a neutral average (due to the sample size being too small) is different from the simple average result. Is this possible?
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39 views

Comparing Binomial R.V.s to Poisson and Normal Approximations

I'm hoping someone can clarify this problem I was given, because I'm not quite sure if my steps are correct. Q: Compute P(X <= 10) when X is a binomial random variable with parameters n = 100, p = ...