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.

learn more… | top users | synonyms

6
votes
1answer
496 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 ...
6
votes
1answer
127 views

Calculating the average of a possibly infinite “compound” length

Sorry for the ambiguous title, I couldn't find a good word to describe my problem. So here is my problem: You are a player, and you have a dice. You have N number of throws available then you can't ...
6
votes
2answers
104 views

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 ...
6
votes
1answer
112 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 ...
6
votes
1answer
966 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 ...
6
votes
2answers
198 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 ...
6
votes
1answer
588 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 $A \ne B$ ...
6
votes
1answer
88 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 ...
6
votes
1answer
131 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 ...
6
votes
2answers
394 views

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 ...
6
votes
1answer
181 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) ...
6
votes
2answers
409 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) ...
6
votes
1answer
118 views

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 = ...
6
votes
1answer
194 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 $\mathbb{E}[W W^{T}]$ ...
6
votes
1answer
676 views

“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 ...
6
votes
0answers
66 views

Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
6
votes
0answers
85 views

Finding an upper bound for $\frac{d}{d\theta}\beta^*(\theta)|_{\theta=\theta_0}$

Suppose that a random variable X has a distribution depending on a parameter $\theta$, $\theta \in \Theta$, and consider a test of hypothesis $H_0: \theta = \theta_0$ versus the alternative $H_1: ...
6
votes
1answer
79 views

About cutting Almonds

Every year, during Christmas baking, I chop almonds, which causes me to puzzle over the same question, and I don't quite know how to approach it. I start out with N almonds. Let's assume they are all ...
6
votes
0answers
523 views

Idempotence and the Rao–Blackwell theorem

Original question: In the Wikipedia article on the Rao–Blackwell theorem, we read: In case the sufficient statistic is also a complete statistic, i.e., one which "admits no unbiased ...
5
votes
5answers
1k views

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 ...
5
votes
4answers
5k views

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 ...
5
votes
3answers
3k views

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?
5
votes
2answers
183 views

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.
5
votes
2answers
322 views

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 ...
5
votes
3answers
803 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 ...
5
votes
4answers
2k 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 ...
5
votes
2answers
4k views

Definition of mean as an integral over the CDF

I'm reading a statistics textbook which defines the mean of a random variable $X$ with CDF $F$ as a statistical function $t(\centerdot)$, where $$ t(F) = \int x \, dF(x).$$ Can someone explain this ...
5
votes
2answers
193 views

Is there a name for the matrix $X(X^tX)^{-1}X^{t}$?

In my work, I have repeatedly stumbled across the matrix (with a generic matrix $X$ of dimensions $m\times n$ with $m>n$ given) $\Lambda=X(X^tX)^{-1}X^{t}$. It can be characterized by the ...
5
votes
2answers
170 views

Improper integral of $\frac{x}{e^{x}+1}$

The improper integral of $\frac{x}{e^x-1}$ (along the positive real line) comes up in a lot of places, you can even invoke the Riemann-zeta and Gamma functions to solve it nicely. However, I just ...
5
votes
1answer
213 views

$X$, $Y$ gaussian variables, $\mathbb{E}[X^2Y]$ and $\mathbb{E}[X^3Y]$ as a function of its means, variances and covariance?

Let be X and Y two not independent Gaussian random variables of means $\mu_X$, $\mu_Y$ and variances $\sigma_X$, $\sigma_Y$, respectively. Let also be $\Sigma$ the covariance between X and Y. I'd ...
5
votes
3answers
382 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 ...
5
votes
5answers
584 views

What does it mean to do MLE with a continuous variable

I am struggling with the semantics of continuous random variables. For example, we do maximum likelihood estimation, in which we try to find the parameter $\theta$ which, for some observed data $D$, ...
5
votes
3answers
156 views

reason of the definition of the covariance

The covariance of two random variables $X$ and $Y$ is defined to be $${\rm Cov}(X,Y) = E[(X-E[X])(Y-E[Y])]. $$ I don't understand it, if someone could explain me this please. Why does this value ...
5
votes
2answers
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 ...
5
votes
2answers
248 views

About joint probability divided by the product of the probabilities

Let $X$ and $Y$ be two events. So $P(X)$ is the probability of $X$ happens, and $P(Y)$ is the probability of $Y$ happens. So $P(X,Y)$ is probability of both $X$ and $Y$ happen. So what is the ...
5
votes
6answers
4k views

What does -1.13 times faster mean?

I'm reading High Performance JavaScript, and I think the graphs in one chapter are just plain wrong. Here is one on Google Books. The y axis is "Times faster", and it runs from -1.5 to +4.0. Now, I ...
5
votes
2answers
2k views

Why is median age a better statistic than mean age?

If you look at Wolfram Alpha or this Wikipedia page List of countries by median age Clearly median seems to be the statistic of choice when it comes to ages. I am not able to explain to myself ...
5
votes
3answers
275 views

Statistics Workshop for High School Students

We are going to hold an introductory workshop about the statistics. The participants will be students who have just finished their 8th or 9th grade. The workshop consists of 10 two-hour sessions. The ...
5
votes
4answers
1k views

Question about basic strategy in Blackjack

I was watching Beating Blackjack with Andy Bloch where he runs through the basic strategy charts that outline the best strategy with playing the game. Later he also talks about the methodologies to ...
5
votes
2answers
245 views

Recommended Reading on Regression Analysis?

For a university project, I am implementing an automated regression analysis tool. However, I have very little background in statistics. So what books / articles / material would you suggest I could ...
5
votes
1answer
237 views

How do you estimate the mean of a Poisson distribution from data?

I have thought of three different approaches for estimating the mean for a Poisson, but I am not sure which one is the correct method to estimate it (the third one is documented separately at the end ...
5
votes
6answers
3k views

Fewest number of moves to win the game 2048?

I'm trying to figure out the fewest number of moves one could make to win the game 2048. In another thread, someone placed the figure at 520, but I'm wondering if anyone knows how to mathematically ...
5
votes
3answers
161 views

How to Make a PDF 'Look' Uniform?

Let $X$ be a normally-distributed random variable with mean zero and variance $\sigma^2$: $X \sim N(0,\sigma^2)$. Let $Y$ be a mapping from $X$ onto the interval $(0,1)$ using the sigmoid function: ...
5
votes
5answers
3k views

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 ...
5
votes
2answers
138 views

Convergence in probability

If $X_1, X_2, \ldots$ converge in probability to a constant $c$, then does $1-X_1, 1-X_2, \ldots$ converge in probability to $1-c$? Is there a way to show this is true / is there an already existent ...
5
votes
1answer
4k views

Probability of duplicate GUID

A GUID (globally unique identifier) is a 32 character hexadecimal string: http://en.wikipedia.org/wiki/Globally_Unique_Identifier If you randomly generate 2, the chance of them being the same is ...
5
votes
4answers
551 views

How to find unique multisets of n naturals of a given domain and their numbers?

Let's say I have numbers each taken in a set $A$ of $n$ consecutive naturals, I ask myself : how can I found what are all the unique multisets, which could be created with $k$ elements of this set ...
5
votes
1answer
126 views

Determining a consistent estimator/asymptotic relative efficiency

Question: Let $X_1,\ldots,X_n$ be i.i.d. as $N(0,\sigma^2)$. a) Show that $\delta_1 = k \sum_{i=1}^n |X_i|/n$ is a consistent estimator of $\sigma$ if and only if $ k = \sqrt{\pi/2}$. b) Determine ...
5
votes
1answer
130 views

Estimating a gaussian distribution from a GMM

Suppose that we have a Gaussian mixture model (GMM) in n-dimensional space: $$P_1(x) = \sum_{i=1}^{C}\pi(c_i)\mathcal{N}(\mu_i,\Sigma_i)$$ We want to estimate a single Gaussian distribution from ...
5
votes
2answers
117 views

Functions whose derivatives can be written as a function of themself?

What kinds of function $f: \mathbb{R} \to \mathbb{R}$ can be written as some function of itself? I.e. $f'(x) = g(f(x))$ for some function $g$? If $f$ is given, can $g$ be solved in terms of the ...