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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Why “bother” with a null hypothesis at all?

(note: this is a very basic probability question, so it is highly probable (heh) that it is a duplicate) Every time I am trying to get into statistics (again), I am always lost at hypothesis testing. ...
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325 views

Efficient computation of $E\left[\left(1+X_1+\cdots+X_n\right)^{-1}\right]$ with $(X_i)$ independent Bernoulli with varying parameter

Suppose we have the random variables $X_1, \ldots, X_n$ that have Bernoulli distributions with the (possibly different) probabilities $p_1, \ldots, p_n$. For example, $X_1$ = 1 with probability $p_1$ ...
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Is there a simple test for uniform distributions?

I have a function that (more or less) is supposed to select a small number $m$ of random numbers from the range $[1,n]$ (for some $n \gg m$) and I need to test that it work. Is there an easy to ...
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1answer
243 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 ...
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655 views

Fast computation/estimation of the nuclear norm of a matrix

The nuclear norm of a matrix is defined as the sum of its singular values, as given by the Singular Value Decomposition of the matrix itself. It is of central importance in Signal Processing and ...
12
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1answer
97 views

Is this condition enough to determine a random variable?

For two positive random variables X,Y, we know that if $E[X^r]=E[Y^r]$ for any r holds, the cdf of X and Y can still be different. Then it occur to me what about changing $r\in \mathbb{N}$ to $r\in ...
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Is there a proof of Benford's Law? [duplicate]

As stated by Wikipedia (here): Benford's law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed ...
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Examples of Simpson's Paradox

I'm looking for fresh examples of Simpson's paradox for use in my statistics courses. The examples I've been using are fine, but I'd like to have some new ones, and I'm hoping folks here might know a ...
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Zero correlation does not imply independence

I just learned that when discussing variables, although independence implies zero correlation zero correlation does not necessarily imply independence. While I understand the concept, I can't imagine ...
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1answer
5k views

sum of squares of dependent gaussian random variables

Ok, so the Chi-Squared distribution with n degrees of freedom is the sum of the squares of n independent gaussian random variables. The trouble is, my gaussian random variables are not independent. ...
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781 views

You see a route 14 bus on the moon. What is the most likely number of bus routes on the moon?

This question was asked on a forum and while many argued that the answer is 14 (since the probability of you seeing bus 14 is maximum in this case), I argued against it that they were working ...
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273 views

What is the difference between probability and statistics?

Is it that probability is top-down (going from pure distributions to predictions about events) and statistics is bottom-up (going from specific events to predicting pure distributions?) I'm pretty ...
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493 views

How many books are in a library?

My cousin is at elementary school and every week is given a book by his teacher. He then reads it and returns it in time to get another one the next week. After a while we started noticing that he ...
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1answer
507 views

Math Intuition and Natural Motivation Behind t-Student Distribution

I am trying to understand with basic mathematical background how the $t$-Student distribution is a "natural" $pdf$ to define. So I hope that this not too-general a question, but given that the ...
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1answer
402 views

Hyper Birthday Paradox?

There are $N$ buckets. Each second we add one new ball to a random bucket - so at $t=k$, there are a total of $k$ balls collectively in the buckets. At $t=1$, we expect that at least one bucket ...
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Multiplicative version of Mcdiarmid's inequality?

Suppose you have $n$ i.i.d. random variables taking values in $\{0,1\}$, and $X$ represents their sum. Then you can use a Chernoff bound to control the deviation of $X$ from its expectation. The ...
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How many times to roll a die before getting two consecutive sixes?

Basically, on average, how many times do you have to roll a fair six-sided die before getting two consecutive sixes?
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Is there a closed form expression for $\int_{- \infty}^\infty \int_{-\infty}^y \frac{1}{2 \pi} e^{-(1/2) ( x^2+y^2 )} \mathrm{d}x\,\mathrm{d}y$?

I have been trying to evaluate the integral: $$\int_{- \infty}^\infty \int_{-\infty}^y \frac{1}{2 \pi} e^{-(1/2) ( x^2+y^2 )}\mathrm {d}x\,\mathrm{d}y$$ I know of course that the integral equals ...
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What is degree of freedom in statistics?

In statistics, degree of freedom is widely used in regression analysis, ANOVA and so on. But, what is degree of freedom ? Wikipedia said that The number of degrees of freedom is the number ...
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1answer
417 views

M.SE reputation distribution

What distribution does the reputation points per user follow on math.SE (or on entire stackexchange)? Is there a mathematical explanation/model of it?
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32k views

maximum estimator method more known as MLE of a uniform distribution

Let $ X_1, ... X_n $ a sample of independent random variables with uniform distribution $(0,$$ \theta $$ ) $ Find a $ $$ \widehat\theta $$ $ estimator for theta using the maximun estimator ...
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Statistics: Could someone show why this exponential pdf integrates into this particular cdf

I have the following exponential distribution: $$f(\lambda, x) = \begin{cases} \lambda e^{-\lambda x} &\text{if } x \geq 0 \\ 0 & \text{if } x<0. \end{cases}$$ I need to show that this ...
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What is the use of moments in statistics

Can any one give an "simple" explaination about what is the use of moments in statistics.Why we need moments? what we can learn from it? if possible please use less equations. Advance thanks for your ...
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With $n$ balls and $n$ bins, what is the probability that exactly $k$ bins have exactly $1$ ball?

I've got a balls and bins problem. Suppose I throw $n$ balls uniformly at random into $n$ bins. What is the probability that exactly $k$ bins end up with exactly $1$ ball? I know this seems a ...
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442 views

Does variance do any good to gambling game makers?

People always like to evaluate the variance, but is there any way for variance to be interesting to the gambling game makers? In another word, what is a pratical gambling game that involving some ...
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The pseudoness of pseudorandom number generators

Is there a reasonable statistic test one can do to standard random number generators (say, one of those that come built in in Python libs) which shows they are not really random? (By reasonable I ...
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1answer
4k views

easy to implement method to fit a power function (regression)

I want to fit to a dataset a power function ($y=Ax^B$). What is the best and easiest method to do this. I need the $A$ and $B$ parameters too. I'm using in general financial data in my project, which ...
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530 views

Expected Value of R squared

Let $n$ be a fixed positive integer. Generate $n$ numbers $x_1, x_2, ..., x_n$ from the set $[0,1]$, with the probability distribution being the uniform one and the $x_i$ all being independent of each ...
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1answer
13k views

comparing distribution of two data sets

I need to compare the distribution (unknown) of a set of data to the distribution of another one (unknown). In particular, I want to check for equality of the two distributions. What are some ...
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2answers
1k views

What is the relationship between the Boltzmann distribution and information theory?

I'm reading a paper on Boltzmann machines (a type of neural network in Machine Learning), and it mentions that "The Boltzmann distribution has some beautiful mathematical properties and it is ...
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286 views

How can you trust a bettor?

Two friends of mine are trying to convince me that they are good at betting. They infact have a certain profit after n bets. They provided me with all the bets they have made so far: odds, outcome and ...
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220 views

Finding a more direct way to reach $\mathbb{E} \left( \sum (X_i - \mu)^2 \right) - \mathbb{E} \left( \sum (X_i - \overline{X})^2 \right) = \sigma^2$

Let $X_i$ be independent random variables, $\forall\,i \in \mathbf{n} \equiv \{0,\dots,n-1\}$, with identical expectation value $\mathbb{E}(X_i)=\mu$, and identical variance ...
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1answer
217 views

Random walks in $\mathbb{Z}^2$

Consider a random walk on the integer lattice in the plane. If a “particle” making a random walk arrives at a lattice point $p = (k_1,k_2)$ at the time $t$, then one of the four neighbors $(k_1±1, k_2 ...
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1answer
125 views

Random point distribtion

How to generate numerically a set of random points $(x_1,y_1), (x_2,y_2),\cdots, (x_N,y_N)$ such that the pair-wise distances $d = \sqrt { (x_i-x_j)^2 + (y_i-y_j)^2}$, for all $ 0<i\le N, ...
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Looking for references related to an inequality in order statistics

I was reading the paper "on the minimum of several random variables". In example 10 item (ii) it states: Let $1\leq k\leq n$. Let $g_i,i\leq n$, be independent $N(0,1)$ Gaussian random variables. ...
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2answers
1k views

Why is polynomial regression considered a kind of linear regression?

Why is polynomial regression considered a kind of linear regression? This is what I mean by polynomial regression. For example, the hypothesis function is $$h(x; t_0, t_1, t_2) = t_0 + t_1 x + t_2 ...
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What does expected value of sum of two discrete random variables mean?

I am confused with summing two random variables. Suppose $X$ and $Y$ are two random variables denoting how much is gained from each two games. If two games are played together, we can gain $E[X] + ...
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What is the deepest / most interesting known connection between Trigonometry and Statistics?

I'm teaching both at the same time to different classes in high school, so I just wondered about this. Added by OP on 16.May.2011 (Beijing time) I mean Statistics only, without Probability. In ...
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Intuition about the Central Limit Theorem

I'm studying statistics, and would like to better understand the Central Limit Theorem. The proof I found on Wikipedia requires some previous knowledge I do not currently possess. Is there a quick ...
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7k views

How to calculate relative error when true value is zero?

How do I calculate relative error when the true value is zero? Say I have $x_{true} = 0$ and $x_{test}$. If I define relative error as: $\text{relative error} = \frac{x_{true}-x_{test}}{x_{true}}$ ...
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4answers
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How would I figure out how many anagrams of mississippi don't contain the word psi?

I'm really confused how I'd calculate this. I know it's the number of permutations of mississippi minus the number of permutations that contain psi, but considering there's repetitions of those ...
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4answers
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A good book on Statistical Inference?

Anyone can suggest me one or more good books on Statistical Inference (estimators, UMVU estimators, hypotesis testing, UMP test, interval estimators, ANOVA one-way and two-way...) based on rigorous ...
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6answers
1k views

Recommend a statistics fundamentals book

To give you some background, I have a grasp on the basics of statistics and probability theory and even remember touching Bayes theorem at the university data mining course. But being a few years away ...
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3answers
190 views

How to win Matt Parker's jackpot - finding the median of the following distribution

In a recent video the legendary Matt Parker claimed he kept flipping a two-sided (fair) coin untill he scored a sequence of ten consecutive 'switch flips', i.e. letting $T$ denote a tail and $H$ a ...
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3answers
1k views

In what sense is the Jeffreys prior invariant?

I've been trying to understand the motivation for the use of the Jeffreys prior in Bayesian statistics. Most texts I've read online make some comment to the effect that the Jeffreys prior is ...
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1answer
128 views

On the central limit theorem

The Central Limit Theorem states for a sequence of i.i.d. random variables $\{X_i\}$, $$\frac{\overline{X} - \mu}{\sigma/\sqrt{n}} \to N(0,1)$$ in distribution as $n \to \infty$. I saw in some ...
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176 views

Is Entropy = Information circular or trivial?

I have seen several "maximum entropy distributions" used in the mathematical and statistical literature, often with the justification that they are "minimally informed" beyond the assumptions and data ...
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413 views

How far do I need to drive to find an empty parking spot?

A parking lot consists of an infinite row of bays. Cars arrive at random intervals (mean interval $T_a$) and stay for a random time (mean stay $T_s$). The time intervals are memoryless (negative ...
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1answer
351 views

On the probability of obtaining 27 sets in the card game Set

For people not familiar with the card game Set, see its entry on Wikipedia and/or one of the related questions here on Math SE. It might be faster to just play the game a couple of times though, see ...
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885 views

Maximum likelihood and sufficient statistics

$$f_T(t;B,C) = \frac{\exp(-t/C)-\exp(-t/B)}{C-B}$$ where our mean is $C+B$ and $t>0$. so far i have found my log likelihood functions and differentiated them as follows: $$dl/dB = ...