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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17 views

Relative Error $\frac{x-x_0}{x}$

According to many definitions I've seen the relative error is defined by $$E = \frac{x-x_0}{x}$$ where $x$ is the "true" value. But some people use instead $$\frac{x-x_0}{x_0}. $$ Is this incorrect? ...
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7 views

Confusion about the definition of Limiting Variance.

On page 470 of Casella and Berger's "Statistical Inference" (2nd edition), they define limiting variance as: For an estimator $T_n$, if $\lim_{n \to \infty} k_n$ Var $(T_n) = > \tau^2 < ...
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22 views

How to find the right way to pick a random selection

I love the website AnyDice (http://anydice.com). When I needed to make a dice game, it saved me hours of coding and testing. Now, I need something similar, but not for dice. I'm trying to do a ...
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1answer
34 views

Chebyshev Application [closed]

If we have a sample mean of $\overline{X} = (X_1 + X_2+\ldots+ X_n)/n$ and mean $m$ and standard deviation $s$, how large should the sample size $n$ be so that with probability $.99$ the error ...
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1answer
20 views

Checking if pseudorandom numbers fit a normal distribution.

I don't have much background in statistics, and one of the exercises in my programming course asks for the following: Generate a sample of normally distributed data using rejection sampling and ...
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38 views

Exponential distribution unbiased estimator

Let $$X_1, \ldots, X_n \overset{iid}{\sim} Exp(\lambda), \quad \lambda > 0$$ The Maximum-Likelihood-Estimator is given by $$\widehat{\lambda} = \frac{1}{\frac{1}{n}\sum_{i=1}^{n}{X_i}} = ...
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3answers
69 views

Probability of winning a dice game.

There is a game with two players and they throw a pair of dice. Player 1 wins if the sum is 7 and player 2 wins if the sum is 6. The stakes are split if neither player wins. What is the expectation of ...
3
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1answer
52 views

The Largest Gap

If I have an X amount of randomly generated positive numbers, what type of algorithm could I run to find the following: Precisely where the largest difference exists between two of the numbers? How ...
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1
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46 views

Expected value of $X$ given $X > Y$

I'm doing some research and I'm trying to compute a closed form for $ \mathbb{E}[ X \mid X > Y] $ where $X$, $Y$ are independent normal (but not identical) random variables. Is this known?
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7 views

Standard Deviation of 2D vector data

Given a sample set of wind data (speed and direction parallel to the earth), I would like to identify the consistency of wind samples. Standard deviation comes to mind, but I don't know if it is ...
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17 views

Can a rectangular pulse be understood as a normal distributed pulse?

Can a rectangular pulse be understood as a pulse with a normal distribution, like a gaussian pulse? In signal-processing there are some specific properties for gaussian pulses. I wondered if you can ...
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8 views

multivariate statistics

I am writing scientific paper(it has been rejected),and I am trying to explain multivariate statistical analysis.I have cited one book'multivariate statistical analysis is concerned with data ...
2
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1answer
36 views

Does the concept of “dynamic average” makes any sense?

While making an excel table about how many times an event happens per day I thought that it could be interesting to see what is the growth rate of those events. If in $2$ days the event happens two ...
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22 views

Statistics one-sided 90% confidence interval

The mean weight of $168$ neonates in the control group of a study on sepsis was $1109$ grams with $s=269 g$. Find what’s called a one-sided $90\%$ confidence interval for the population mean: a value ...
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46 views

Does this equation have a solution?

Suppose $Q:[0,1]^2\to R$ and $f,g:[0,1]\to R_+$ are probability densities. $$\int_{[0,1]}[P(x,y)-Q(x,y)]f(x)dx=C_1, \forall y$$ $$\int_{[0,1]}[Q(x,y)-P(x,y)]g(y)dy=C_2, \forall x$$ s.t. ...
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33 views

Can we ever have E(argmin(f)) = argmin(E(f))?

Consider a parametric real-valued function $f_{\boldsymbol{\alpha}}:\ \mathbb D^N \rightarrow\mathbb R$ whose parameters $\boldsymbol\alpha$ vary according to some distribution $\psi$, and $\mathbb D$ ...
0
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1answer
44 views

conditional probability that 5 red balls were placed in the bowl at random

Place five similar balls (each either red or blue) in a bowl at random as follows: A coin is flipped 5 independent times ad a red ball is placed in the bowl for each head and a blue ball for each ...
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1answer
14 views

Determining the value of ECDF at a point using Matlab

I have a data $X=[x_1,\dots,x_n].$ In Matlab, I know by using [f,x]=ecdf(X) plot(x,f) we will have the empirical distribution function based on $X$. Now, if ...
3
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1answer
54 views

fantasy basketball model

i'm creating a fantasy basketball model (could be used in other games too) where we can project how well a player will do against another team even when the player hasn't played against a certain team ...
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0answers
10 views

Weighted Average of Correlation Matrix

Let $R$ and $Q$ be two correlation matrices of the same size and let $p\in[0,1]$. I'm trying to show that $pR+(1-p)Q$ is still a correlation matrix. I claim that $\sqrt pX+\sqrt{1-p}Y$ is a vector ...
6
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1answer
81 views

How Do I Find My Car

I have been discussing this problem with a coworker for a few days now and neither of us have made any headway on it. I would appreciate any help with a possible solution or maybe a suggestion of a ...
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0answers
16 views

Expectation of a function of Gamma random variable

Consider a truncated exponential distribution $F(x\left| \lambda \right.) = \frac{{ - {e^{ - \lambda x}} + {e^{ - \lambda }}}}{{ - {e^{ - 2x}} + {e^{ - \lambda }}}}$ on the interval $[1,2]$. The ...
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17 views

Can a randomized rule induce a random measure on the action space?

$D = \{d_i: X\to Y, i=1,\dots,n\}$ is a finite set of mappings from $X$ to $Y$, $(\Omega, \mathcal F, P)$ is a probability space, and $\delta: \Omega \to D$ is a measurable mapping. Can $\delta$ ...
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1answer
20 views

Sum of normally distibuted random variables?

I feel terribly confused over this. If: X~N(μx,σx^2) Y~N(μy,σy^2) both independent random variables Z=X+Y Z~N(μx+μy,σx^2+σy^2) Then why: Xi~N(μi,σi^2) i=1, 2, ..., n X1, X2, ..., Xn are ...
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68 views

Given mean and standard deviation for weight of CDs, find the average weight for boxes holding CDs

A box holds 100 CDs. Average weight of a CDs is 1 ounce with a standard deviation of 0.05 ounces. The box is considered overweight if it weighs more than 100.4 ounces. 400 boxes are shipped. a) What ...
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14 views

Cartesian Product converted into Summation

I am looking at the proof of Maximum Likelihood Estimator and So let's get to it: first take the $\log$ of the equation: $$\log(P(\text{DATA}))=\log\prod i=1N(PX_i(1−P)1−X_i)$$ Since $$\log(a ...
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10 views

Correct choice of analytical statistic method for time series problem

I want to make a statement about corruption (y) influenced by the ratification of the UN-convention-Against-Corruption "contract" (x). Luckily, most of them signed it within 3 years. So, I have ...
0
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1answer
12 views

PDF of uniform distribution

Find the uniform distribution of the continuous type that has he same mean and the same variance as those pf a chi--square distribution with 8 degrees of freedom. My solution: for the mean- $\frac ...
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6 views

Asymptotic Variance of estimator of partial slope of endogenous variable from IV reg

Suppose we have a linear model $y=Q+Rx+error$, where $E(error)=0$, and $z$ is an instrument for $x$ (endogenous) where the correlation between the instrument and the error is 0 but that between the ...
2
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1answer
30 views

probability matching strategy for coin flips

imagine a betting game where we observe $N$ independent coin flips $x_1,...,x_n$ (where each $x_i \in {H,T}$) from the same coin, whose true weight is $\theta$. the task is to predict how many Heads ...
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1answer
28 views

Solve Integral $ \int_{-\infty}^{+\infty} \log[F(y_k)]*F(y_k)^{\frac{1-\theta}{\theta}}*f(y_k) $ for MVUE proof

Let $ \theta > 0 $ a parameter, $ Y_1 ... Y_n $ is a set of iid observations with marginal distribution function $F_\theta(y) = [F(y)]^{1/\theta} ,-\infty < y < +\infty $ Show that $ t = ...
4
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1answer
66 views

delta-method-ish question

This is a problem from Resnick's heavy tail analysis book: Let $\{X_n\}$ be a sequence of random variables such that $EX_n=m$ and Var$(X_n)=\sigma_n^2>0$ for all $n$, where $\sigma_n^2\rightarrow ...
0
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1answer
36 views

Power of hypoteses test

I am having some problems calculating the power in this exercise. http://snag.gy/LkulQ.jpg This is what i've tried. So since i know that $\alpha$ is 0.05. I thereby know that the upper boundary ...
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2answers
24 views

resolving expected utility of st. petersburg paradox with logarithmic utility

St. Petersburg paradox is a game where you toss a fair coin repeatedly and if it lands heads on the $k$th trial you get $2^n$ dollars. Expected utility of game is: $E(U) = \sum_{k=1}^{\infty}[0.5*0 + ...
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0answers
22 views

Hypothesis test for means or variances

I construct a permutation test to test whether A,B come from the same distribution. Now I want to test if either means or variances are different. How can I construct a hypothesis test to detect if ...
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1answer
26 views

Expectation of multinomial distribution

Three fair dice are cast. In 10 independent casts, let X be the number of times all three faces are alike and let Y be the number of times only two faces are alike. Find the joint pdf of X and Y and ...
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23 views

How to compare different theoretical models with the experimental model?

I have experimental model, which is represented by the real function $f_e(E)$. This function was computed by the set of experimental points, which are unknown for me. I have developed theoretical ...
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1answer
51 views

What does it mean to have a lone plus sign in the exponent/superscript (Modified Weiszfeld algorithm)

I was skimming through Vardi & Zhang's paper on "A modified Weiszfeld algorithm for the Fermat-Weber location problem". It is available at http://www.stat.rutgers.edu/home/cunhui/papers/43.pdf ...
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14 views

Test whether the proportion of days which do not have “Good” air quality has changed.

I have the following data of AQ-value $$36,52,54,60,60,31,22,13,67,59,52,54,30,12,33,39,50,51,34,56,62,51,33,39,23,12,35,28,41,54,44,66,42,33,28,22,36,57,46,39etc.$$ I have total $364$ such values in ...
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0answers
14 views

Find the solutions to this equation involving a normal distribution.

The equation is as follows: http://i.imgur.com/xv2zsZR.gif Some hints and comments: A-Each side is a distribution where sigma is a constant and represents the variance. B-Using simple plotting ...
0
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0answers
30 views

Hypothesis Test with Unknown Distribution i.e. $X_i \stackrel{iid}{\sim} (\mu, \sigma^2)$

You have the following sample draw X = {34,28,50,102,18,39,90,71,140,19,89,55} of a random variable $X_i$ assume to be iid with finite means and variances. Test $H_o: \mu = 50$ against a one-sided ...
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0answers
26 views

How can I infer order from partially ordered discrete sequences?

A really interesting problem that I can't stop thinking about! Have run in to this a couple of times but yet to find a smart approach to either solve or frame this problem. This is my try at ...
0
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1answer
19 views

Joint PDF Correlation

In the problem I am given $f(x,y)=2,\ 0 < x < y,\ 0 < y <1$. I'm trying to find the correlation $\rho$ which I know is equal to $$\rho = \frac{Cov(x,y)}{\sqrt{Var(x)Var(y)}}$$ ...
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46 views

IB HL Mathematics IA Exploration - Benford's Law

As the title suggests, I am writing a paper on Benford's Law as part of my High School curriculum. I have read some of the other stack exchange proofs for Benford's law, and explained benford's law ...
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21 views

Rao-Cramer lower bound regularity condition and dominated convergence

Let $(\mathcal{X}, \mathcal{F}, (\mathbb{P}_\vartheta)_{\vartheta \in \Theta})$ be a statistical model dominated by a sigma-finite measure $\mu$ with Likelihood-function $L(\vartheta, x)$ which is ...
0
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1answer
25 views

Neyman-Pearson lemma and hypothesis test

Let $X_1...X_n$ be idndependent and identically distributed as Bernoulli random quantities with parameter $p$. Let $p_1>p_0$ and consider the test of hypotheses $H_0:p=p_0$ versus $H_1:p=p_1$ ...
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11 views

Describing statistical outliers in specification data sheets

I have an instrument that is interrogated (say) once per second to provide a data value. After reading it for (say) 1000 seconds I have a good idea of how well it performs, with Average and Standard ...
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1answer
12 views

Permutation test and p-value

I construct a permutation test in order to see If two samples come from the same distribution or not. I have two vectors $x, y$ that hold values of sampled values from two populations and the test ...