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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How to make statistical sense of this experiment:

I have conducted an experiment but I am now unsure of how to say, from a statistics point of view, that the data supports or not that a certain phenomenon has occurred, meaning it could be mere ...
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19 views

Distribution of the test statistic?

Let $\mathbf{x}_i \sim \mathcal{N}(\boldsymbol\mu, \boldsymbol\Sigma)$. I am trying to find a distribution of the following test statistic $ T(\mathbf{x}) = \frac{\bar{\mathbf{x}}^H ...
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Meaningful Extreme value distribution

Extreme value theory (EVT) dictates that the limit distribution of the minimum of the set of i.i.d. Chi-square random varibales $\{C_1,C_2,\cdots,C_n\}$ is Weibull. The Weibull distribution has ...
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14 views

An optimization problem for non-homogenous poisson process with unknow distribution

Jobs arrive at an M/M/1 type server according to an non-homogenous Poisson process with rate parameter $\lambda_k$. Where $\lambda_k$ and $\mu_k$ denotes the arrival rate and service rate at $k_{th}$ ...
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1answer
18 views

Which is the probability?

On the Swedish SAT test, you have 5 options for every question where precisely one option is correct. If you answer randomly, what is the probability that your score will be 0.9 if the maximum score ...
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28 views

Probability that one normal Random Variable will fall within a given range of another.

I'm struggling with the following problem: (ed: Don't be lazy. Just type it out. ) A certain small freight elevator has a max. capacity $C$, which is Normally distributed, with mean ...
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13 views

Help in calculating the Hessian Matrix from the log-likelihood

I am trying to find the Fisher Information Matrix for a univariate linear linear Moving Average model: \begin{align} z(n) &= h_1 u(n-1) + h_2 u(n-2) + u(n) \tag{1} \\ y(n) &= \mathbf{h^Tz(n)} ...
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1answer
38 views

How to summarize a big table of results? Average or Geometric mean?

I am writing a paper for a Computer Science conference and I have a big (way too big) table of results (times and some other measures) for different versions of an algorithm. I would like to summarize ...
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17 views

In statistics using a regression analysis in SPSS - - variables are hunger and amount of dancing

In statistics using a regression analysis in SPSS - - variables are hunger and amount of dancing. Which would be the dependent and independent variables? Thanks!
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24 views

Distribution of a product of Multinomials

Consider the following: $(X_1, X_2, X_3, X_4) \sim \mathrm{Multinomial} (n,\mathbf{p})$ where $\mathbf{p} = (p_1,p_2,p_3,p_4)$. I would like to find the distribution of $X_1 X_4$, or at least know ...
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31 views

Definition of standard deviation and $l_2$

If we denote the mean as $\mu$, then the standard deviation is: $$\sigma\equiv\left(\sum_{x\in X}{p(x)(x-\mu)^2}\right)^\frac{1}{2}$$ In other words, $\sigma$ is the average $l_2$ distance from $\mu$. ...
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How to take into account uncertainty on number of events

Suppose I generate a set of events $X_{i}$ for $i = 1,2 \dots N$ and suppose every event is either a success or a failure, ie. $X_{i} = 0, 1$. If $N$ is fixed, the MLE for the probability of success ...
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1answer
30 views

Applications of statistics to pure mathematics [on hold]

Are there any "applications" of statistical methods to pure mathematics?
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1answer
39 views

A related problem regarding Normal Distribution (Continuous Probability) [on hold]

A circus performer who gets shot from a cannon is supposed to land in a safety net positioned at the other end of the arena. The distance he travels is normally distributed with a mean of 140 feet and ...
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1answer
38 views

Coin Toss Experiment

I conducted an experiment where I tossed a coin 100 times. I am assuming that the coin flips heads with a probability p=0.5. So that the coin is fair with a level of significance of 5%, I want to find ...
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53 views

Binomial distribution, given the number of success, what is the expected total number of trials?

For a random variable that follows binomial distribution, $X|N=n\sim Binomial(n,p)$. What is the expectation of $N$ when we know the value of the random variable but don't know the total? ie. What is ...
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13 views

Statistics Sampling Type

My question is on Q7. I can't seem to figure this one out. I thought it was a random statified cluster sample because it is breaking down the schools into subsections and then pulling 3 homerooms ...
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11 views

Uncertainty of calibrated Bevel Protractor

I have one problem regarding to ''Uncertainty of Universal Bevel Protractor''. I want to verify a External certificate which is calibrated from outside Lab. Can 'Calibration Uncertainty' be bigger ...
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24 views

Central Limit Theorem Absolute Inequality Equation

$\mu_N = 0.5$ (mean) $\sigma_N^2= \dfrac{1}{12N}$ (variance) $\sigma_N = \sqrt{\dfrac{1}{12N}}$ (standard deviation) For $N = 100$, what is $P(|X_N - (1/2)| > 0.025) =$ ? (i.e.using Central ...
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12 views

Proving convergence of inverse covariance matrix (aka precision matrix)

Let $\Sigma$ be the population covariance matrix and $\hat{\Sigma}$ be the sample covariance matrix. It is well known that $\hat{\Sigma} \rightarrow \Sigma$ in the large sample limit. I have also ...
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1answer
38 views

Continuous distribution and independence [on hold]

Problem: In a room, there are 4 boys from high income families, 6 girls from high income families and 6 boys from low income families. How many girls from low income families also need to be present ...
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Of strings and substrings: A problem of probability

Problem Let $\Sigma=\{a, b\}$. Let $\Sigma^*$ denote the Kleene star of $\Sigma$: \begin{equation*} \Sigma^* = \{\varepsilon, a, b, aa, ab, ba, bb, aaa, aab, \ldots\} \end{equation*} where ...
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54 views

Probability and continuous distributions

Suppose that the daily consumption of pepsi in ounces is normally distributed with normal(13, 4) in ounces. The daily amount consumed is independent of other days except adjacent days where the ...
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1answer
25 views

Relationship between chi-square goodness-of-fit testing and chi-square distribution?

Anyone care to explain the relationship between a Chi-square goodness of fit test and the Chi-square distribution? One has the expected value in the denominator and the other has the variance in the ...
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1answer
67 views

Stock market trading / Casino betting / Multi-player fun competition possible with the following input? [on hold]

I would like to program some kind of online betting system for fun. Just for the fun factor, I would like the Twitch chat to be the random input (seed). As can be seen here, you can see one possible ...
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1answer
19 views

How are Chi Square probabilities calculated?

What steps would one follow to calculate the values in a Chi Square probability table such as https://people.richland.edu/james/lecture/m170/tbl-chi.html? Say you had 15 degrees of freedom and wanted ...
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1answer
38 views

Probability of histogram bars

Say I collect data that follows a Normal distribution $f(z)$ in a histogram with bins of width $w$. I want to calculate the probability that the number of hits $N_i > N_j$. My naive approach would ...
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Concentration inequalities for product of gaussians

Are there any concentration inequalities (i.e. probability bounds on how a random variable deviates from its expectation) for the product of $n$ gaussian random variables with zero means and equal ...
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1answer
32 views

How to find PDF of ordered random variables? [on hold]

Assumpion: Let $X_1, X_2, \ldots, X_L$ be $L$ independent and identical random variables (RVs). Let $F_{X_i}(x_i)$ and $f_{X_i}(x_i)$ be CDF and PDF of $X_i$. Suppose that $F_{X_i}(x_i) = F_X(x_i)$ ...
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1answer
27 views

Probability Ques. [on hold]

From previous experience, Bob’s Programming teacher takes down the attendance 40% of the time. Bob’s classmate, Marty, comes late to class (i.e. after the attendance is taken down) 20% of the time. ...
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50 views

Confidence Interval for Incidence Rates

I have a huge sample of patients followed up for a certain Event. I would like to calculate the following crude incident rates: #{Events}/(1000*PersonYear). My sample is big enough to assume that this ...
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1answer
42 views

Approximation to a compounded Binomial distribution

I need to find an approximation, from which I can easily sample, to the following compounded Binomial distribution: $X \sim \mathrm{Binomial}(e^{-\epsilon}, \ n)$ where $\epsilon \sim ...
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What is the mode if every value only ocurres once (there are no repeated values) [duplicate]

Say i have these values: 1, 2, 3,4, 5, 6, 7, 8 ... Normally, the mode should be the number that was repeated the most. But here every number appears only once. Are all of them considered as modes? ...
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29 views

How do I calculate conditional PDF?

Obtain $$P(2 < Y < 3 | X = 1)$$ where the joint pdf of X and Y is $$f_{X,Y}(x,y) = (6-x-y)/8$$ where $$0 < x < 2$$ and $$2 < y < 4$$? so first, I did $$f_Y|X=1(y) = ...
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19 views

Find the required Chi-square score for an arbitrarily low p-value (2 degrees of freedom)

I'm trying to use the Chi-Square test to find the significance of data that suffers from the multiple testing problem. Because I have this multiple testing problem, the required p-value to view a test ...
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8 views

RBF transformation on a Normally Distributed Random Variable

I have a random vector $\mathbf{X} \sim \mathcal{N}(\mathbf{m,\Sigma})$ which is transformed by a Gaussian Radial Basis Function into the random variable $\mathbf{Y} = K(\mathbf X)$ where $K = ...
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Departure from uniformity in a continuous (time) distribution

I know how to quantify the departure from uniformity ( or a uniform distribution) for discrete distributions. Assume you have a distribution set of P: ...
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Is Autocorrelation computed with cross-correlation or correlation?

I have seen multiple articles online that say that Autocorrelation is computed using correlation and cross-correlation. Which is autocorrelation actually computed with?
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22 views

Probability of a user references in a network [on hold]

I am trying to figure out no of possible referrals of a user in a network. Where the size of a network is not fixed but we can set an assumption of 1000 persons. Edit: A user knows few users in a ...
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2answers
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statistical significance in probabilities

Imagine I am conducting an experiment, and I record whether $n$ individuals of different nationalities, say $A$, $B$, and $C$, either like or dislike a product. In the end I have the respective ...
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1answer
34 views

Proof of a classical Theorem of Martin-Löf on complexity dips for Kolmogorov complexity,

I have a question on the first Theorem from the article Complexity of Oscillations in Infinite Binary Sequences by P. Martin-Löf, which could be downloaded from the publisher or from here. Theorem ...
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1answer
49 views

Binomial Distribution Confidence Interval

Normally when I see confidence intervals it is in attempt to estimate a population parameter (probably poor wording). What I am trying to do is form a confidence interval for some theoretical values, ...
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1answer
17 views

computing weight from distance metric

I have a distance between two points in meters. I want to convert this distance into weight such that as distance increases the weight decreases. What are some good weighting function that can ...
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1answer
15 views

How to calculate a posterior probability with a given Gaussian Mixture Model?

I'm building a GMM-based classifier in speech processing and I'm using GMM as a probabilistic scoring mechanism (therefore I don't intrinsically care about the underlying mixture components). For ...
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1answer
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How do I solve this question using Z Table and Normal distribution?

A company pays its employees an average wage of 15.90 an hour with a standard deviation of 1.50. Assume the wages are approximately normally distributed. a) what proportion of employees receive ...
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How do I use interpolation with the Z table?

My textbook has an example of interpolation, but I am not sure how the book did it since it doesn't explain it. It says if we want $P(Z < 1.246)$ we must use interpolation and the steps given are: ...
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2answers
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Given a variable $X$ with a PDF, what is the PDF of $\sqrt{X}$

I feel this is simple and I'm overlooking something really basic. Let's say a have a variable $x$ which obeys the exponential distribution. So if collect 100000 occurrences of $x$ and plot its ...
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1answer
32 views

Intuition on Martin-Löf-Test for finite strings

The followng example is from An Introduction to Kolmogorov Complexity and Its Applications, Example 2.4.1. and is concerned with Martin-Löf-Tests for finite strings: A string $x_1 x_2 \ldots x_n$ ...
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1answer
25 views

Show that $Y = \sum_{i=1}^n Y_i$ is distributed as $\chi _{2n}^2$.

The Statement of the Problem: Suppose that $X_1,\ldots, X_n$ is a random sample from the $U(0,1)$ distribution and $$ Y_i = -2\log X_i. $$ Show that $Y = \sum_{i=1}^n Y_i$ is distributed as $\chi ...
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1answer
27 views

How to find median from a probability distribution?

Having trouble on something that should be really, really easy. I need to find the median of the following probability distribution...but according to the website I linked below...I'm doing it ...