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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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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445 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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185 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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153 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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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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224 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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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 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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1k 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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239 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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980 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 ...
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235 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 ...
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182 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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2k 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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152 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: ...
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120 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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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 ...
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460 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 ...
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94 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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115 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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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 $S_n = 3$. ...
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Probability density function of a product of uniform random variables

Let $z = xy$ be a product of two uniform random variables, with $x$ having the range $[a, b)$ and $y$ the range $[c, d)$. What is the probability density function of $z$, and how is it calculated?
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856 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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Statistics and confidence - intervals

An account on server A is more expensive than an account on server B. However, server A is faster. To see whether it's optimal to go with the faster but more expensive server, a manager needs to ...
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pairwise correlation of three random variables

Assume three random variables have all equal pairwise correlation. What are the possible values of this correlation? Can all of these values be achieved? The solution says $\rho \in [-\frac 12,1]$, ...
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Great Book on Probability and Statistics (for Computer Scientists)

I'm a Computer Science sophomore and we're studying Probability and Statistics (fundamentals and all). The teacher recommends a book which I don't like since it does not even try and explain ...
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What is the equation used to calculate a linear trendline?

In excel it is done automatically but how to manually calculate a linear trendline over a set of points?
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4k views

Defective items probability question.

Hi I'm working with probability as part of an engineering course, and I'm struggling with the following tutorial question: Components of a certain type are shipped to a supplier in batches of ten. ...
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318 views

Ways of getting a number with $n$ dice, each with $k$ sides

Assume the dice are numbered from $1$ to $k$. My hunch is that this will form a normal distribution with a median at $n\cdot\frac{k}{2}$. However, I have no idea as to turn this fact into an answer ...
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503 views

What is the relationship between the Poisson Distribution and the Monte Carlo Fallacy?

Gravity's Rainbow has this long passage about the Poisson distribution. Since Pynchon's education included a serious dose of mathematics, and his novels include many references to mathematics, I ...
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X,Y are independent standard normal distributed then what is the distribution of $\frac{X}{X+Y}$

X, Y are independent standard normal random variables, what is the distribution of $$ \frac{X}{X+Y} $$ Could anyone help me with this? Thanks. I have worked the problem by multivariable ...
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556 views

Probability of Monkey typing keyboard

A monkey types at a 26-letter keyboard with one key corresponding to each of the lower-case English letters. Each keystroke is chosen independently and uniformly at random from the 26 possibilities. ...
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486 views

Expected Value of a Continuous Random Variable

I've been reviewing my probability and statistics book and just got up to continuous distributions. The book defines the expected value of a continuous random variable as: $E[H(X)] = ...
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301 views

If $X_1, …, X_n$ are Exp($\lambda$) random variables, what is the best unbiased estimator of $e^{-\lambda}$?

Let $X_1, ..., X_n$ be random variables with pdf $$\frac 1 \lambda e^{-x / \lambda} I(x > 0).$$ The goal is to find the best unbiased estimator of $h(\lambda) = e^{-\lambda}$ (incidentally, this ...
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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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What is the difference between all types of Markov Chains?

I have been looking for some good material covering Markov Chains but everything seems so difficult to me... After reading about the subject, I figured out that there is basically three kinds of ...
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395 views

expectation of $ \left(\sum_{i=1}^n {x_i} \right)^2 $

If $x_i$ is exponentially distributed $(i=1,...,n)$ with parameter $\lambda$ and $x_i$'s are mutually independent, what is the expectation of $\left(\sum_{i=1}^n {x_i} \right)^2$ in terms of $n$ and ...
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Calculating a sample size based on a confidence level

It's been a while since my last statistics class... I have 404 files that went through some automated generation process. I would like to manually verify some of them to make sure that their data is ...
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46 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 ...
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64 views

$X-Y$ equivalent in distribution to $0$?

If $X$ is equal to $Y$ in distribution, is it equivalent to $X-Y$ which is equivalent in distribution to $0$?
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366 views

The Birthday Problem

I've been reading about the birthday problem which, as I'm sure many of you will know, is a statistical problem which aims at finding out the how many people you would need in a random group to be ...
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124 views

Highest points in heart cycle graph

I'm making an application that reads the heart cycle from a device, and I've aimed to get this image: Now, I need to get the highest points that appear in every cycle in order to calculate the ...
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269 views

Central Limit Theorem Definition

My friend and I have a bet going about the definition of the Central Limit Theorem. If we define an example as a number drawn at random from some probability density function where the function has a ...
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1answer
173 views

Conditioning on an event with probability close to one

Let $(\Omega,\mathcal{F},P)$ be a probability space. If $A\in\cal F$ is an event with $P(A)=1$, then $$ P_{\mid A}(B)=P(B\mid A)=\frac{P(B\cap A)}{P(A)}=P(B),\quad B\in\cal F. $$ I wonder if something ...
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Probability/Combinatorics Problem. A closet containing n pairs of shoes.

A closet contains n pairs of shoes. If 2r shoes are chosen at random, (where 2r < n), what is the probability that the chosen shoes will contain no matching pair? I have tried thinking about this ...
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Greatest common denominator of measurements

In a couple months, I'll do the Millikan experiment. Then, I'll end up with a number of charge measurements and their errors $$((q_i, \Delta q_i))_{i \in \mathbb N}.$$ The idea is that all those $q_i$ ...
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145 views

Proof of $ \text{Var}\,\left(\sum_{i=1}^{n}g(X_i)\right)=n\left(\text{Var}\,g(X_1)\right).$

I have a question about part of a proof of a Lemma in a book (Casella's Statistical Inference) I'm reading. This it how it goes. Let $X_1, \cdots ,X_n$ are a random sample from a population and ...
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187 views

Finding the joint distribution of $X_{1:n}$ and $\overline{X}$

I need to show that, given a random sample of independent variables $X_1, ... , X_n$, each following a distribution EXP($\theta$,$\eta$), that is, ...
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1answer
113 views

Expected overlap

Suppose I have an interval of length $x$ and I want to drop $n$ sticks of unit length onto it (where $\sqrt x<n<x$). What is the expected overlap between sticks? ($x$ can be assumed to be large ...
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1answer
201 views

Statistics formula for wifi positioning.

Assuming I have $3$ access point namely: $AC_1$, $AC_2$ and $AC_3$ and I want to know my location using this access point and a mobile device that will get signal from the access points. First thing ...