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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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
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1answer
791 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 ...
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0answers
55 views

Justify an unbiased estimator is UMVUE

Suppose $X_1,\ldots,X_n$ are iid $N(\theta,\theta)$, with $\theta\in(0,\infty)$. Is $\bar{X}$ the UMVUE (beta unbiased estimator) of $\theta$? I find the complete sufficient statistic is ...
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29 views

Find a function such that follows to normal in distribution

Suppose that $X_{n}\sim \text{Binomial}(n,\theta)$, where $n=1,2,\ldots$ and $0<\theta<1$. Find a function $g$ such that $\sqrt{n}(g(\frac{1}{n}X_n)-g(\theta))\xrightarrow{D} N(0,1)$ for each ...
6
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1answer
104 views

What is a formal definition of 'randomness'?

What is a rigorous mathematical/logical definition of 'randomness'? Under what conditions can we truthfully apply the predicate 'is random'?
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66 views

An estimator for the c.d.f $F$ at a point $x_0$?

Problem: Let $X_1,X_2,\ldots,X_n$ be independent identically distributed random variables (i.i.d's) with common CDF $F$. Fix $x_0\in\mathbb{R}$ and find an unbiased estimator for $F(x_0)$. Show ...
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82 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 $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
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100 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
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1answer
91 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 ...
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5answers
2k 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 ...
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4answers
7k 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 ...
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3answers
4k 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?
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11k views

Sum of two independent geometric random variables

Let X and Y be independent random variables, $ P(X = k) = P(Y = k) = p(1 - p)^{k-1} $ How do you show that the pmf of $ Z = X + Y $, is negative binomial, and how do you find $ P(X = Y) $?
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2answers
191 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.
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2answers
348 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 ...
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3answers
944 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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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 ...
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5answers
9k views

Given a data set, how do you do a sinusoidal regression on paper? What are the equations, algorithms?

Most regressions are easy. Trivial once you know how to do it. Most of them involve substitutions which transform the data into a linear regression. But I have yet to figure out how to do a ...
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1answer
13k views

What is the expectation of $ X^2$ where $ X$ is distributed normally?

I know that if $X$ were distributed as a standard normal, then $X^2$ would be distributed as chi-squared, and hence have expectation $1$, but I'm not sure about for a general normal. Thanks
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2answers
203 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 ...
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2answers
174 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 ...
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5answers
704 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 $D$, ...
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1answer
250 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 ...
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3answers
421 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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4k 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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5answers
31k views

If I roll two fair dice, the probability that I would get at least one 6 would be…

11 out of 36? I got this by writing down the number of possible outcomes (36) and then counting how many of the pairs had a 6 in them (1,6) (2,6) (3,6) (4,6) (5,6) (6,6) (6,5) (6,4) (6,3) (6,2) (6,1). ...
5
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3answers
159 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 ...
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3answers
3k views

Expected value of the product of functions of two independent random variables

If $X$ and $Y$ are independent random variables, are the statements below true $$E(e^{X+Y} ) = E(e^X)\times E(e^Y)$$ and $$E(X^2\times Y^2) = E(X^2)\times E(Y^2)$$ , where $E(\cdot)$ = expectation? ...
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288 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 ...
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4k views

Finding expected number of distinct values selected from a set of integers

I have a set of $n$ integers $\{1, . . . , n\}$, and I select three values with replacement. How can I find the expected number of distinct values? Note each value is chosen uniformly and ...
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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 ...
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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 ...
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3answers
3k 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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3answers
362 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 ...
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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
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2answers
257 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
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1answer
322 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 ...
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6answers
4k 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 ...
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3answers
3k views

Improbable vs Impossible?

I was wondering how mathematics in general or any of its sub fields e.g.statistics, probability, define the words Improbable and Impossible. I get their English meaning, that something is impossible ...
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3answers
169 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
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1answer
37k views

How to determine if Standard Deviation is high/low

I have derived the following response time data for a performance test I am running: Min - 8sec Max - 284sec Average - 28sec Standard Deviation - 27sec What does the standard deviation say about the ...
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3answers
3k views

Example of Sufficient and Insufficient Statistic?

I am having trouble understanding the concept of a sufficient statistic. I have read What is a sufficient statistic? and Sufficient Statistic (Wikipedia) Can someone please give an example of: a ...
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4answers
2k views

Estimate probabilities from its moments

I want to estimate probability $Pr(X \leq a)$, where $X$ is a continuous random variable and $a$ is given, only based on some moments of $X$ (e.g., the first four moments, but without knowing its ...
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2answers
155 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 ...
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4answers
700 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
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1answer
159 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 ...
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2answers
4k views

Deriving Moment Generating Function of the Negative Binomial?

My textbook did the derivation for the binomial distribution, but omitted the derivations for the Negative Binomial Distribution. I know it is supposed to be similar to the Geometric, but it is not ...
5
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1answer
163 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 ...
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2answers
119 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 ...
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3answers
90 views

Bernoulli Random Variables and Variance

The question is: Suppose $Z_1, Z_2, \ldots $ are iid $\operatorname{Bernoulli}\left(\frac{1}{2}\right)$ and let $S_n = Z_1 + \ldots +Z_n$. Let $T$ denote the smallest $n$ such that $S_n = 3$. ...