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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What is the difference between $\mathbf X\mathbf X^\prime$ and $(\mathbf X-\mu)(\mathbf X-\mu)^\prime$?

In $\mathbf X\mathbf X^\prime$, $\mathbf X$ is a matrix contains data points in column fashion, $\mathbf X^\prime$ is its transpose, this looks like a covariance matrix, but does not subtract mean, so ...
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259 views

What's the proper function to calculate confidence percentages for Chi Squares?

In the book Practical BASIC Programs by Lon Poole (Osborne/McGraw-Hill, 1980) (~10M pdf), the sample program listing has a number of discrepancies and some outright incorrect statements. On page 143 ...
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0answers
535 views

Use of covariance matrix for the confidence interval

I have a number of explanatory variables $x_1,...,x_n$ and an outcome variable $y = f(x_1,...,x_n)$. Here $f$ is assumed to be known (estimated). I heard that for a confidence interval for $y$ one can ...
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1answer
582 views

What's the probability that a bookmaker is mispricing odds on soccer games

An English soccer team plays a series of matches again different opponents, of varying ability. A bookmaker offers odds for each match as to whether it will be a home win, away win, or draw. Part-way ...
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249 views

Can I build a program that will tell me if a real world data set looks linear, logarithmic, exponential etc?

I have a bunch of real world data sets and from manually plotting some of the data in graphs, I've discovered some data sets look pretty much logarithmic and some look linear, or exponential (and some ...
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279 views

The Metropolis Algorithm

I Know how to apply the Metropolis Algorithm, but I'd be grateful if someone could explain to me the reasoning behind the steps in the algorithm. I've tried in vain looking for the original paper. ...
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1answer
107 views

Determining the distribution of a population from a sample

I have a uniformly collected sample of 10000 data points from a population of about 200000. I'd like to find out what the distribution of the population is. How can I do this rigourously?
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370 views

Usefulness of Variance

I've had a look for intuitive explanations of the variance of an RV (e.g. Intuitive explanation of variance and moment in Probability.) but unfortunately for me, I still don't feel comfortable with ...
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505 views

Time Until Extinction for a Pure Death Process Where the Time is Exponentially Distributed

Let $X(t)$ be a pure death process starting from $X(0)=N$. Assume that the death parameters are $\mu_1, \mu_2,\dots,\mu_N$. Let $T$ be an independent exponentially distributed random variable with ...
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548 views

How to compare randomness of two sets of data?

Given two sets of random numbers, is it possible to say that one set of random numbers has a greater degree of randomness when compared to the other? Or one set of numbers is more random when compared ...
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1answer
90 views

Probability and Z-Scores

I'm new here and was hoping you guys could help me with a statistics problem that I don't quite understand. I'm not sure if it's proper etiquette to ask for help on a specific homework problem here, ...
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115 views

What are the conditions under which a distribution reduces down a dimension?

What are the conditions under which a distribution reduces down a dimension? For example, suppose I have a 2D gaussian distribution for X and Y. Under what condition(s) on Y does the distribution ...
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1answer
62 views

Variance of a recursed substition function

So, I have this function: $$ y_t=v_t+\rho v_{t-1}+\rho^2 v_{t-2}+\dots+\rho^{t-1}v_1+\rho^ty_0 $$ And I want to find the variance (and after that the covariance, but I should be able to do that..). ...
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779 views

Can you demystify the Power Law?

How would you describe the Power Law in simple words? The Wikipedia entry is too long and verbose. I would like to understand the concept of the power law and how and why it shows up everywhere. For ...
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1answer
1k views

leading and lagging moving average indicator

What are leading short and lagging long moving average indicators and how do we calculate them? e.g. for the following data set, and let window size be 2. Can you ...
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1answer
132 views

posterior distribution after having partial information on some linear combinations of unknown variables (Revised)

$x_1$, $x_2$, and $x_3$ are i.i.d. normal random variables with distribution $N(0, \sigma_x^{2})$ $\epsilon_1$, $\epsilon_2$, and $\epsilon_3$ are i.i.d. normal random variables with distribution ...
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1answer
173 views

How does the median change when the underlying variable changes

Suppose that a random variable $X \in [0,1]$ is drawn according to the density $f(x|\theta)$ conditional on the realization of $\theta \in [0,1]$. From $x$, we can generate $c$ according to some rule ...
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3answers
201 views

What is the statistically natural way to center the logit-scale around a given value?

I would like to find out the formula for CandidateAbility used in the European PISA-test, which tests 9th grade pupil's abilities. Unfortunately the agency which publishes the results does not provide ...
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3answers
822 views

The relationship between mean and variance in the context of system energy and the partition function

I'm looking at a specific derivation on wikipedia relevant to statistical mechanics and I don't understand a step. $$ Z = \sum_s{e^{-\beta E_s}} $$ $Z$ (the partition function) encodes information ...
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3k views

Characteristic function of the normal distribution

The standard normal distribution $$f(x) = \frac{1}{\sqrt{2\pi}} e^{\frac{-x^2}{2}},$$ has the characteristic function $$\int_{-\infty}^\infty f(x) e^{itx} dx = e^{-\frac{t^2}{2}}$$ and this can be ...
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463 views

How to calculate $E [Z^2]$ when $E[Z]$ is known?

For the following calculation, why $E[Z_1^2] = \pi/4$ ? In general, how to calculate $E[Z^2]$ ? Thanks much. 3 down vote Let's also elaborate on Ross Millikan's answer, adapted to the case ...
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135 views

component and dimension in Gaussian mixture model

What is the relation between a dimension and a component in a Gaussian Mixture Model? And what is the meaning of dimension and component? Thank you. Please correct me if I'm wrong: my understanding ...
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2answers
477 views

Chebyshev's inequality, variance and mean

I am trying to implement a solution (working code) for the 4.1 paragraph in this paper. The problem: We have words with lengths for instance: $l_1$ = 1, $l_2$ = 2, $l_3$ = 3, $l_4$ = 8 and $l_5$ = ...
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1answer
222 views

Extension of a monotone likelihood ratio property (revised)

Suppose the conditional distribution of $\theta$ and $x$ satisfies the monotone likelihood ratio property: for every real numbers $\bar{\theta} \ge \theta$ and $\bar{x} \ge x$, we have $ ...
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Proof of upper-tail inequality for standard normal distribution

$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$, $$ \mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>. $$
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1answer
3k views

Sample sizes for an infinite population

I've poked about in some other questions, and I'm no sure how to deal with my problem and my knowledge of statistics has atrophied. Particularly that I'm trying to choose a sample size for a ...
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1answer
142 views

Variance of $\overline{X}_n^2$

Here is a problem a have got in my homework. Given a set of $X_1, ... X_n \sim F$ i.i.d values find the variance of $T_n = \overline{X}_n^2$ where $\overline{X}_n = \frac{\sum_i{X_i}}{n}$. I actualy ...
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1answer
348 views

Total Variation in Probability Intuition

Let $\Omega$ be a finite probability space and define the Total Variation as $TV(\mu,\nu):=\sup\{|\mu(A)-\nu(A)|: A\in\Omega\}$. One can also derive that ...
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4answers
138 views

Finding the optimum supply quantity when there is uncertainty in forecast

This is actually a quiz that will be needed in a real life food stall! I need to decide how much stock to supply for my pumpkin soup stall. I sell each soup for $5$ dollars a cup, and let's say my ...
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1answer
720 views

Simple stats question-correlation coefficient

Let's say we have two exams, each out of 50 points. The correlation rate between them is 0.75. If the teacher decides to add 10 points to the results of the first test, what will happen to the corr. ...
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7k views

How to calculate odds

Tim hortons has a roll-up the rim to win contents. your chances of winning in a single play is 1 in 6. so you have a 5/6 (83% chance) of winning nothing. How do I calculate the odds of losing ...
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834 views

How to calculate a confidence interval for a binomial, given a specific prior

I'm trying to come up with a 95% confidence interval for the click-through-rate of particular advertisement. It has $x$ clicks out of $n$ impressions so far. What's the best way to compute this, ...
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294 views

Does this calculation have a name, or a generic formulation?

Background I would appreciate help in identifying / explaining this operation: To calculate each of the $n$ values of $f(\Phi)$: sample from the distribution of each of $i$ parameters, $\phi_i$ ...
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4answers
1k views

Geometric mean never exceeds arithmetic mean

This was a mathematical induction question proposed in a textbook, and I've exhausted multiple approaches (proving RHS - LHS > 0, splitting the fraction, fractional exponents, etc.) The geometric ...
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1answer
3k views

Proving correlation coefficient = 1 or -1 given X and Y=a + bX

Given $X$ and $Y = a+bX$, I have to prove that: If $b \lt 0$, then $\rho = -1$. If $b \gt 0$, then $\rho = 1$. I've gotten to the point where I have: $$ \rho = \frac{b \cdot \sigma_x }{ ...
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1answer
441 views

Ti-83 Probability of events occurring

First time here, more frequent over at StackOverflow. So not sure if calculator questions are accepted here, but I'll ask. Is there a quick way to solve problems like this in the ti-83: 82% shot ...
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2answers
223 views

dividing samples in equal slabs

I have sample data like in above format along X, Y axis. Now what i would like to do is to devide it in "n" number of slabs having fixed values. Now how do i achive this in mathematics(statistics). ...
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1answer
5k views

How to calculate the expected value $E(XY)$ with known $E(X)$, $E(Y)$ and $\sigma_{i}$?

I am trying to understand the value of $\bar{x_{1} x_{2}}+E(x_{1} x_{2})$. For all $i$, $E(x_{i})$ and $\sigma_{i}$ are given. Wikipedia gives the joint probability density function: $E(XY) = \int ...
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1answer
442 views

Monotonic behavior of a function

I have the following problem related to a statistics question: Prove that the function defined for $x\ge 1, y\ge 1$, ...
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1answer
241 views

A question about F-distribution

Let $f(m,n,w)$ be the probability density function of F variable with m numerator df and n denominator df, i.e. ...
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1answer
737 views

Questions about geometric distribution

I have some trouble understanding the record value for a sequence of i.i.d. random variables of geometric distribution. Following quotation is from Univariate discrete distributions By Norman Lloyd ...
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1answer
80 views

Stats help please!

Being a lit. student who doesn't even really know what all the math signs are, I have a question. It is probably super easy for anyone with q math iq higher than my room-temperature one. The scores ...
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2answers
972 views

Good Online Resources to learn Multivariate Statistics?

Are there any good online resources to learn multivariate statistics? ( With topics such as Multiple Linear Regression, Principal Component Analysis, Factor Analysis, Cluster Analysis and ...
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1answer
223 views

Maximum Likelihood notation

My notes for determining the Maximum Likelihood decision rule for classifying two fish given two features (lightness and width) have a different decision rule if the features are correlated to when ...
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1answer
194 views

Probability from a collection of independent predictions

Reframed question: Team A and Team B will be playing each other in a sporting event where there is one winner and one loser; a tie is impossible. A man, wanting to make money from placing a bet on ...
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2answers
534 views

How to calculate a confidence interval for $p$, the parameter of the binomial distribution?

Suppose I have a random variable X that is known to follow the binomial distribution B, but whose parameter $p$ is unknown. I have observed 100 samples of X, and they all came out true. How can I ...
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1answer
524 views

Use a Trendline Formula to Get Values for Any Given X with Excel

First off, is this question appropriate for this Stack Exchange? If not, I'll remove it. Now to my question: Is there an easy way to apply the trend line formula from a chart to any given X value in ...
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148 views

Understanding some basics

Let $x_1,\ldots,x_n$ be any numbers and $\bar{x} = (x_1+ \ldots + x_n)/n$. Then I need to prove that $$min_a \sum_{i=1}^n (x_i - a)^2 = \sum_{i=1}^n (x_i - \bar{x})^2$$ I do not need the answer but ...
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7k views

how does expectation maximization work?

I'm reading a tutorial on expectation maximization which gives an example of a coin flipping experiment (the description is at ...
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209 views

Expectancy value for the percentage of points lying in the Convex Hull (3D)

Suppose I chose n uniformly distributed random points in a 3D cube. What is the expected value for the percentage of points lying on the convex hull as a function of n? Just as a reference, I made ...