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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Is there any difference between statistical learning and machine learning?

Straight to the point, I'm a math student and I have a course this year called Statistical Learning. From the description, the course contains: Large datasets analysis, regression, principal ...
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0answers
14 views

Weighting the data by the history

I have a input stream 3D data that comes every time frame. Each point is defined by 3D vector of x,y,z. There is a evaluation function [say f(x)] that computes if the point at time t is valid or ...
2
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1answer
36 views

Cramer-Rao lower bound for normal($\theta, 4\theta^2$)

I am trying to find the Cramer-Rao lower bound for unbiased estimators of $\theta$, given a sample $X_1,\ldots, X_n \sim \textrm{normal}(\theta,4\theta^2)$. I am calculating the CRLB as $$ ...
2
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1answer
54 views

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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1answer
24 views

Normal distribution calculations

We have a gaussian distribution $$ X \sim N(\mu,\sigma^2)$$ where $\mu = 4$ and $\sigma^2 =1.5$ . Probability is given by : $P(x<c)=0.35$ $c$ needs to be calculated. And we got ...
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2answers
29 views

Class Coin Toss Experiment

My classmates and I are doing a coin toss experiment (i.e. toss coin 100 times). I have already determined that I have a fair coin, since I tossed $43$ heads, and this falls into a $95$% confidence ...
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2answers
32 views

What is the variance of the volumes of particles?

According to Zimmels (1983), the sizes of particles used in sedimentation experiments often have a uniform distribution. In sedimentation involving mixtures of particles of various sizes, the larger ...
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2answers
23 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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1answer
28 views

Pairing birthdays [on hold]

How large a group of people would you need to provide a better than 50-50 chance that everyone will have at least one birthday (just month/day) partner?
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21 views

What is the proof behind the mean confidence interval for a Binomial Distribution?

How do we obtain the range to be as [$\mu-$$zσ$, $\mu+$$zσ$]? Is it when $n$ is sufficiently big?
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2answers
60 views

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
53 views

Coin Toss Experiment

I conducted an experiment where I tossed a coin $n=100$ 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 ...
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22 views

How Kriging, Bochner theorem and Positive definite (PD) function are related?

This question referes to the link: https://en.wikipedia.org/wiki/Kriging I can understand the relation between Bochner's theorem and PD function. But could not properly understand and connect all ...
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1answer
12 views

Extreme value distributions of unaccountably infinite set of random variables

Let us suppose that we have an uncountably infinite set $A=\{x_1,x_2, \cdots\}$ of i.i.d. random variables $x_i$, say with gamma distribution. Are minimum and maximum extreme value distributions ...
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1answer
38 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
30 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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1answer
40 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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14 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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0answers
24 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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0answers
12 views

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 ...
2
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0answers
54 views

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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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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0answers
14 views

Justification for Inverse-Variance Weighting?

In Inverse-Variance Weighting, $1/\sigma^2$ is defined as the weight of the random variable. Why is this?
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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
18 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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2answers
27 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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2answers
18 views

Statistics and data analysis

What is meant by confidence interval in data analysis e.g. 95% confidence interval? How does p<0.05 estimate significant difference?
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1answer
32 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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1answer
14 views

Finding out number of observation

There are $n$ scores $X_1,X_2,X_3,....,X_n$ and their sum is $80$ and sum of their squares is $400$ then which among them is the probable value of $n$ A)$10$ B)$9$ C)$15$ ...
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1answer
34 views

Applications of statistics to pure mathematics [on hold]

Are there any "applications" of statistical methods to pure mathematics?
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1answer
45 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 ...
3
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4answers
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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1answer
1k views

Statistics - Z-score confusion.

For z score, you are taking the sample value subtracting population mean and dividing it by std deviation. Is that correct so far? Now, the "sample value" is defined by an equation. In my scenario, I ...
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0answers
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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0answers
13 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 ...
3
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1answer
295 views

Sufficient conditions for convergence of functions of random variables

I hope this question is not too general, but I am not completely sure yet how to phrase it. So we all know that when we have two sequences of random variables $X_n$ and $Y_n$ for $n \ge 1$, that ...
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1answer
25 views

How to find total numbers of people with height higher than some value?

Is a continuous distribution. The y axis is not probability, how do I find how many people with height higher than some value? Now assume it is an even distribution. y is no. of people, x is ...
1
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1answer
39 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 ...
2
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1answer
18 views

The distributions of incomes in two cities follow the two Pareto type pdfs. Find P(X<Y)

The distributions of incomes in two cities follow the two Pareto type pdfs $$f(x)= \frac{2}{x^3}, 1 < x < \infty.$$ $$g(y) = \frac{3}{y^4}, 1<y<\infty.$$ Here one unit represents ...
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0answers
55 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 ...
0
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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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0answers
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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0answers
19 views

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 ...
3
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1answer
69 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 ...
32
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8answers
7k views

Why do we use a Least Squares fit?

I've been wondering for a while now if there's any deep mathematical or statistical significance to finding the line that minimizes the square of the errors between the line and the data points. If ...
2
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1answer
43 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 ...
2
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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 ...
2
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5answers
14k 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
34 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 $F_{X_i}(x_i) = F_X(x_i)$ ...
2
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0answers
140 views

Building Bayesian Networks, Causality and Cyclic Reasoning

I am studying Bayesian Statistics and I am trying to get a good understanding on Bayesian Networks, which seems to be vital in order to make something useful in Machine Learning. Most of the texts I ...