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 accurately calculate the error function erf(x) with a computer?

I am looking for an accurate algorithm to calculate the error function I have tried using [this formula] (http://stackoverflow.com/a/457805) (Handbook of Mathematical Functions, formula ...
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103 views

If $X \sim N(0,1)$, why is $E(X^2)=1$?

If $X$ is a normally distributed with mean $0$ and variance $1$, expectation of $X$ equals $0$ but why is $E(X^2)=1$?
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What happens if I toss a coin with decreasing probability to get a head?

Yesterday night, while I was trying to sleep, I found myself stuck with a simple statistics problem. Let's imagine we have a "magical coin", which is completely identical to a normal coin but for a ...
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Proof of the independence of the sample mean and sample variance

I've been trying to establish that the sample mean and the sample variance are independent. One motivation is to try and write the sample variance, $S^{2}$ as a function of $\left\{ ...
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What does it mean when a statistician says I’m 90% confident that the mean of the population is between 1 and 9?

Does that mean if I draw samples from the population that 90% of the time I'll get a number between 1 and 9? Added: assume normal distribution for the population.
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What can I do with measure theory that I can't with probability and statistics

I've studied mathematics and statistics at undergraduate level and am pretty happy with the main concepts. However, I've come across measure theory several times, and I know it is a basis for ...
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What is the expected number of runs of same color in a standard deck of cards?

Standard deck has $52$ cards, $26$ Red and $26$ Black. A run is a maximum contiguous block of cards, which has the same color. Eg. $(R,B,R,B,...,R,B)$ has $52$ runs. $(R,R,R,...,R,B,B,B,...,B)$ has ...
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How was the normal distribution derived?

Abraham de Moivre, when he came up with this formula, had to assure that the points of inflection were exactly one standard deviation away from the center, and so that it was bell-shaped, as well as ...
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What is the T-distribution, and what is it used for?

(I'll post my own answer to this, but don't hesitate to post your own!) Student's t-distribution, or T-distribution, was introduced in 1908 by William Sealey Gossett writing under the pseudonym ...
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Birthday “Paradox” - another, different, version!

Background Many people are familiar with the so-called Birthday "Paradox" that, in a room of $23$ people, there is a better than $50/50$ chance that two of them will share the same birthday. In its ...
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What is a good measure of “controversy”, given a support score and opposition score?

Suppose I have a topic or discussion, and a number of "support" and "opposition" points on each side (You can also think of them as "upvotes" and "downvotes") and I want to calculate a score of how ...
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What is the expected number of dice one needs to roll to get 1,2,3,4,5,6 in order?

If I have a fair die and throw it until I get a run of 1,2,3,4,5,6 in order, how many times on average must I throw the dice?
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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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727 views

Expected number of card draws to get all 4 suits

You have a standard 52 card deck, with 13 cards of each of the 4 suits (Hearts, Diamonds, Spades, Clubs). What is the expected number of cards you have to draw from the deck until you have all 4 suits ...
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219 views

Is there a way to check the correctness of your answer to a probability question?

In CS, there's a systematic way to check if your code is buggy or not as you write code. Is there a way to check the correctness of your answer to a probability question without using a textbook? For ...
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325 views

Function of a random variable: expectation

Let $\{X_i\}_{i=1}^n$ be a sequence of i.i.d. random variables (i.e. a random sample) with pdf: $$f_X(x) = e^{-(x-\theta)} \, e^{-e^{-(x-\theta)}} · \mathbf{1}_{x\in \mathbf{R}}$$ The goal is ...
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How to find a confidence interval for a Maximum Likelihood Estimate

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 was ...
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117 views

Monte-Carlo for the Wasserstein metric

Let $(X,d)$ be some metric space and assume that $d\leq 1$. Further, let $\mu, $ $\nu$ be two Borel probability measures on $X$ and let $$ \Gamma(\mu,\nu) = \{\gamma - \text{measure on }X\times ...
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253 views

What is a confidence interval?

What are the nature and purpose of confidence intervals?
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888 views

The probability of a drunk person/random walk

A drunk person wonders aimlessly along a path by going forward 1 step and backward 1 step with equal probabilities of $\frac12$. a) After 10 steps, what is the probability that he has moved 2 steps ...
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Is There Something Called a Weighted Median?

I was given some data that represents the number of lines in a document as well as the line count per hour (which is the lines in the document divided by the number of hours that the document was ...
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Bag with infinite number of colored balls

Consider a situation with a bag with infinity number of balls. Each ball is of some color. Number of colors is finite but it is not known. Balls are drawn from the bag one by one and checked for the ...
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446 views

Statistics: Why do we divide by $\sqrt{n}$ for sample standard deviation

Can someone tell me if my explanations/understanding is on the right track? Suppose we have a set of variances, each of them identical, where $V_{1}(x) + V_{2}(x) + ...+ V_{j}(x) = \sigma^2$. If we ...
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Why are additional constraint and penalty term equivalent in ridge regression?

Tikhonov regularization (or ridge regression) adds a constraint that $\|\beta\|^2$, the $L^2$-norm of the parameter vector, is not greater than a given value (say $c$). Equivalently, it may solve ...
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Average percent increase not equal to total percent increase?

I tried searching around for this but it was difficult to boil down the search terms. Plus nothing seemed to be showing up anyway. What's an easy way to show that the average percentage increase of n ...
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Spinners from yesteryear: A challenging probability problem

While browsing the Internet I found an old horse racing game where the results were determined by a spinner. The names of 6 different horses were listed an equal number of times on the spinner. Each ...
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101 views

Asymptotic efficiency of maximum likelihood estimate

Let us consider a simple statistical model $\{f_{\theta}\}$ where $\theta\in U$, an open subset of $\mathbb{R}$. Let $X_1,\dots,X_n$ be sample drawn from $f_{\theta}$. I know, under some regularity ...
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829 views

Maximum Likelihood Estimation of an Ornstein-Uhlenbeck process

I am wondering whether an analytical expression of the maximum likelihood estimates of an Ornstein-Uhlenbeck process is available. The setup is the following: Consider a one-dimensional ...
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195 views

Might such a sequence of mathematical expectations be able to predict uncertain events?

This question might sound a little bit mystical, but it seemed like an interesting idea, so I am posting it here. Despite the title, I know it probably does not work miracles, but here goes anyway. I ...
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548 views

Symmetric matrix decomposition with orthonormal basis of non-eigenvectors

I like to understand the following transformation found in documentation for deriving Kalman filter. Abstract Formulation: Given 2 symmetric matrices $A$ ,$B$ $\in$ $\mathbb R^{3,3}$ with $A \ne B$ ...
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66 views

Seasonal adjustment and Fourier analysis

I've been reading up on seasonal adjustment (removing "seasonal" periodic components from a time series) recently and although I see a lot of fancy work around ARIMA models and fancy ways to detect ...
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115 views

How to determine if binomial events are independent?

I have a sequence of binary experiment results, something like 1100010000100... My first hypothesis is that these events are independent, but I'd like to know if there is some way to test this. I ...
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Taylor series approximation statistics

how can I show the following: Let $X_1, X_2,\ldots, X_n$ be i.i.d Poisson with mean $\lambda$. Let $Y = |\{i: X_i =0\}|$. Then $\lambda$ is estimated by $$\eta = - \log(Y/n)$$ Use Taylor series to ...
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180 views

How to find a flat using game theory?

I had the idea that maybe probability/game theory knowledge helps finding a flat more systematically. I assume that I have some online offers with number parameters: prize size (square meters) ...
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324 views

Intuition of Gamma Family

The function $$f(t) = \frac{t^{\alpha-1}e^{-t}}{\Gamma(\alpha)}, \ \ 0 < t < \infty$$ is a pdf. But Why is the gamma family defined as $$f(x| \alpha, \beta) = \frac{1}{\Gamma(\alpha) ...
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Help with convergence in distribution

$Y$ is a random variable with $$M(t) = \frac{1}{(2-\exp(t))^s}.$$ Does $$\frac{Y-E(Y)}{\sqrt{\operatorname{Var}(Y)}}$$ converge in distribution as $s$ tends to infinity? I let $Z = ...
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190 views

If W is a random matrix with variance $\mathbb{E}[W W^{T}]$, what's $\mathbb{E}[W^{T} P W]$?

I know quite a few identities about quadratic forms of random vectors, but I'm having difficulty coaxing something out of this quadratic form of random matrices. Suppose I know $\mathbb{E}[W W^{T}]$ ...
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“How many public playgrounds exist in the United States?” How to answer using statistics and probability

I have a goal of estimating how many public playgrounds exist in the United States. There are many methods of gathering real data about playgrounds, but, unfortunately, there is no single authority ...
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Can an event be possible if its probability is zero?

Consider a computer program that generates any random number between 0 and 1(exclusive). There are infinitely many numbers between 0 and 1. So the probability that the random-number generate the same ...
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Unbiased Estimator for a Uniform Variable Support

Let $ x_i $ be iid observations in a sample from a uniform distribution over $ \left[ 0, \theta \right] $. Now I need to estimate $ \theta $ based on $N$ observations and I want the estimator to be ...
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What is the purpose of the standard deviation?

I don't have any knowledge of statistics beyond high school common sense. Why is the standard deviation usually seen in combinatorics textbooks, and why is the standard deviation defined ...
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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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Average wait time arriving at subway randomly

If the subway comes every 10 minutes on average, what is the expected wait time if I arrive at the station randomly? Can someone help me mathematically understand this problem?
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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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How to prove a random variable taking values in $[0,1]$ range has variance no larger than $\frac{1}{4}$?

How can I prove that a random variable taking values in $[0,1]$ has variance no larger than $\frac{1}{4}$? If it matters, discrete and continuous proofs are both welcome.
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Name this paradox about most common first digits in numbers

I remember hearing about a paradox (not a real paradox, more of a surprising oddity) about frequency of the first digit in a random number being most likely 1, second most likely 2, etc. This was for ...
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Proof of $\frac{(n-1)S^2}{\sigma^2} \backsim \chi^2_{n-1}$

It's a standard result that given $X_1,\cdots ,X_n $ random sample from $N(\mu,\sigma^2)$, the random variable $$\frac{(n-1)S^2}{\sigma^2}$$ has a chi-square distribution with $(n-1)$ degrees of ...
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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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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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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 ...