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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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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417 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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432 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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718 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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417 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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Which theory is used to calculate the position and energy of a point source?

Consider an empty room with one point source that emits a stationary signal (constant sound, radioactive radiation, ...). The energy nor the position of the point source is known. We send someone in ...
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144 views

Chances of someone being of a certain gender at websites

I have 2 of websites and I know the chances of a visitor being a female or male. Let's say I have 2 website where the chance of a new visitor being a female is 80%. If the visitor comes on website 1 ...
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599 views

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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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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327 views

Is there an introduction to probability and statistics that balances frequentist and bayesian views?

Perhaps, roughly, I might be described as advanced undergraduate regarding mathematics. However, I have not learned statistics and have only learned elementary probability. Does there exist a book or ...
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280 views

What is a confidence interval?

What are the nature and purpose of confidence intervals?
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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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Given every horse's chance of winning a race, what is the probability that a specific horse will finish in nth place?

I have been interested in calculating a specific horse's chance of finishing in nth place given every horse's chance of winning in a particular race. i.e. Given the following: ...
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sum of two independent exponential distribution

let $Y_1\sim \exp(\lambda_1)$ and $Y_2\sim \exp(\lambda_2)$ and $V=Y_1+Y_2$ Show that the pdf of $p_V(x)$ of $V$ has the following form ...
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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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575 views

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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69 views

distribution of one random over the sum of random variables

Suppose that $X_1,\ldots,X_n$ are independent random variables with $X_i\sim Gamma(\alpha_i,\beta)$. Define $U_i=\frac{X_i}{X_1+\cdots+X_n}$ for $i=1,2,\ldots,n$. Show that $U_i\sim ...
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Linear regression: degrees of freedom of SST, SSR, and RSS

I'm trying to understand the concept of degrees of freedom in the specific case of the three quantities involved in a linear regression solution, i.e. $SST=SSR+SSE, $ i.e. Total sum of squares = ...
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219 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 ...
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Correlated Poisson Distribution

$X_1$ and $X_2$ are discrete stochastic variables. They can both be modeled by a Poisson process with arrival rates $\lambda_1$ and $\lambda_2$ respectively. $X_1$ and $X_2$ have a constant ...
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Correlation Coefficient and Determination Coefficient

I'm really new to linear regression and am trying to teach myself. In my textbook there's a problem that asks why $R^{2}$ in the regression of $Y$ on $X =$ the sample correlation between X and Y the ...
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271 views

Can I build a program that will tell me if a real world data set looks linear, logarithmic, exponential etc?

I have a bunch of real world data sets and from manually plotting some of the data in graphs, I've discovered some data sets look pretty much logarithmic and some look linear, or exponential (and some ...
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Check answer, How to find Cov(x,y) and Var(2x-y)?

I have the following tableau ...
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53 views

Limit of median of uniform distribution

Let $X_1,X_2,\ldots$ be a random sample from the uniform distribution on the interval $(0,1)$. Assuming that $n$ is odd, find the pdf of the sample median (say $M_n$). Does the pdf of the r.v. ...
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118 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 ...
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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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645 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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201 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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673 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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150 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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591 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 ...
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484 views

Why does maximum likelihood estimation work the way that it does?

I'm wrapping my head around MLE right now and there's something about it that bothers me, irrationally I'm sure. I believe I understand the procedure: essentially we hold our observations fixed and ...
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183 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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124 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 = ...
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199 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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Estimate Grade Distribution Based on Performance of Each Question

As the title states, I would like to be able to estimate the grade distribution of an exam based on the mark distribution of each individual question. To give a quick example of what I mean, suppose ...
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75 views

Higher math and statistics/probability

So I've heard that certain areas of statistics and probability use manifolds and results from analysis and topology. Given that I lack the background to see where manifolds would become useful in ...
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31 views

How to tell who is the stronger chess player, for the purpose of fine tuning chess engines? [closed]

This is a chess like question concerning how many games are required to tell who is the stronger player. The application is in fine tuning chess engines. It is typical that during this process a ...
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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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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 ...
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218 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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82 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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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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104 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: ...
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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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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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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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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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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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Expected Value of the maximum of two exponentially distributed random variables

I want to find the expected value of $\text{max}\{X,Y\}$ where $X$ ist $\text{exp}(\lambda)$-distributed and $Y$ ist $\text{exp}(\eta)$-distributed. X and Y are independent. I figured out how to do ...