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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pdf of power of rayleigh variables

I'm trying to compute the pdf of the power of a rayleigh distributed random variable. So, let $X$ be distributed as Rayleigh $$ X \sim \frac{x}{\sigma^2} e^{-\frac{x^2}{2\sigma^2}}, x\geq 0 $$ let ...
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1answer
3k views

The inverse variance is proportional to the sample size?

Interested in meta-analysis I found the lecture notes http://www.uazuay.edu.ec/cibse/lecture_notes.pdf I understand that it makes sense in a meta-analysis to assign different weights to different ...
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265 views

Likelihood Ratio Test

I am having a hard time understanding what this following question is asking. I do not have any clue pn how to start it. Any help would be great. Thanks in advance. Let $X_1,X_2,...,X_n$ be a random ...
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242 views

Scoring system for two-party game

Suppose we have a game that operates with '$a~$' players for one party, and '$b~$' players on the opposing party. Each party has a long-run average win percentage for this game, with $A\%$ and ...
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1answer
307 views

Unbiased estimator and Variance

I am having a hard time trying to solve this problem. I don't know how to start it. Any help would be greatly appreciated. Let T be any unbiased estimator of $\tau(\theta),$ and let $W$ be a ...
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1answer
459 views

convolution of random variables

I need to compute the pdf of the sum of a bunch of random variables $$\sum_{i=0}^{k-1} c_i X_i $$ where $X_i \sim 2\Omega x e^{-\Omega x^2}$, $\Omega > 0$ is a parameter and $c_i$ are positive ...
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1answer
72 views

Getting a single-value estimation of trust in a computed mean

Suppose I have a number N of independent ratings of a given item, where each rating is an integer between 1 and 7 (inclusive). For simplicity sake, let us assume the ratings are normally distributed, ...
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6k views

Is There Something Called a Weighted Median?

I was given some data that represents the number of lines in a document as well as the line count per hour (which is the lines in the document divided by the number of hours that the document was ...
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169 views

calculate the rate of change

I am trying to calculate the change frequency for a set of data. Each bit of data has the date-time it was created. I would like to say for a specific set of data the change frequency is hourly, ...
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2answers
1k views

Percentage of risk, Is my calculation correct?

X is the worst scenario. In order for X to occur, two things must happen, in sequence. An event with an occurrence chance on 1/120 must occur. This is EVENT A. if EVENT A occurs, another event with ...
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2answers
384 views

What relationship between matrices (Multivariate Statistics)

I am working with sampling of multivariate normally distributed numbers. I have a very fundamental question regarding the eigendecomposition of the $k \times k$ covariance matrix $\Sigma = ...
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1answer
3k views

Can median have decimal value for finding missing number?

I was wondering if it was possible to get the median as decimal value for finding the missing median? If there is an even number of numbers in the set, the median is the average of the two middle ...
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701 views

You see a route 14 bus on the moon. What is the most likely number of bus routes on the moon?

This question was asked on a forum and while many argued that the answer is 14 (since the probability of you seeing bus 14 is maximum in this case), I argued against it that they were working ...
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2answers
15k views

maximum estimator method more known as MLE of a uniform distribution

Let $ X_1, ... X_n $ a sample of independent random variables with uniform distribution $(0,$$ \theta $$ ) $ Find a $ $$ \widehat\theta $$ $ estimator for theta using the maximun estimator ...
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2answers
3k views

method of moments of an uniform distribution

Let $ X_1, ... X_n $ a sample of independent random variables with uniform distribution $(0,$$ \theta $$ ) $ Find a $ $$ \widehat\theta $$ $ estimator for theta using the method of moments ...
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1answer
99 views

Convergence of Bayes Error to zero

Given, $$ 0<p_{i} ,q_{i} <1 $$ define ${E}{}_{k}$ as, $$\begin{array}{l} {E_{1} =min(p_{1} ,q_{1} )+min(1-p_{1} ,1-q_{1} )} \\ {E_{2} =min(p_{1} p_{2} ,q_{1} q_{2} )+min(p_{1} (1-p_{2} ...
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3answers
154 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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118 views

order statistics

I would like to understand the relationship between the asymptotic moments of order statistics and the moments of the distribution of the mother distribution. I will appreciate any references on this ...
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1answer
693 views

What is the definition of a “predictable process”?

I am reading a book on financial mathematics, and frequently encounter the phrase "predictable process", which I haven't seen definition of, and cannot find the definition online. At first I thought ...
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1answer
238 views

Calculating Percent Variance Of Dates?

I have this problem where I have a set of data. Each data point has two values: Value1: Birthday Value2: Height Now, the problem I am faced with involves ...
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1answer
227 views

Sufficient statistic for product of $\Gamma(x + a_i)$

Background Suppose that the national mint mints coins with bias $p_i \sim \rm{Beta}(A,B)$ for some unknown constants $A$, $B$. Given $n$ coins, you flip each coin a certain number of times. Coin ...
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2answers
168 views

Statistics reduce

First of all, I'm sorry I'm a complete statistics iliterate, I promise I'm going to get into it after the summer (I need them for my job actually). Because of this I'm not sure about the name of what ...
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124 views

Median of a set of Integer

The median of a set containing odd number of integer is the middle element after sorting. What about the case of even number of terms. Some places mention of average of the two middle value, but I ...
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188 views

monotone likelihood ratio property of truncated distributions

What are the conditions that truncated normal distributions satisfy the monotone likelihood ratio property? How about other truncated distributions?
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1answer
53 views

Any relationship for repeated results for tossed coin?

I tossed coin (fair) 10 times and collected data as follows:- H H T H H T H T H H what I did added "repeat" in observation if previous toss was same; like H (Repeat) H T H (Repeat) H T H T H ...
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1answer
82 views

Solving for Exponents and Logarithms

I was discussing a small experiment with a friend of mine this week. He said, "we can just do 3 trials." I said, "sure, but the subject will have to get all 3 trials right to be better than chance." ...
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1answer
596 views

Magical Loaded Dice and expected values

Let's say we have a magical biased die that allows us to set the probability of every side to whatever we want (nonzero and add up to 1, of course). And let's say this die enforces the condition that ...
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4k views

Proof of the independence of the sample mean and sample variance

I've been trying to establish that the sample mean and the sample variance are independent. One motivation is to try and write the sample variance, $S^{2}$ as a function of $\left\{ ...
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1answer
67 views

deriving a bayesian statistics equation

can someone give me some pointers on deriving p(alpha|rest) on page 4 of http://www.stat.cmu.edu/~brian/724/week06/lec15-mcmc2.pdf (pdf warning)? I'm having trouble separating it out from beta. This ...
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1answer
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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2answers
295 views

Scaling a histogram in the face of outliers

I am trying to figure out how to display a histogram of a digital image in the face of massive outliers (lots of shadows, highlights, or lots of anything inbetween). If I simply choose the bin with ...
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2answers
192 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
760 views

Shortest Confidence interval for $\sigma^2$

I have a $X_1, \cdots , X_n$ random samples from a $N(\mu,\sigma^2)$. $\mu$ is known, but $\sigma^2$ is unknown. I would like to know how to go about constructing a $(1-\alpha)$100% shortest ...
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1answer
738 views

GLR test of hypothesis for exponential distributions

I'm having trouble with this exercise from Bain and Engelhardt's textbook: Consider independent random samples of size $n_1$ and $n_2$ from respective exponential distributions $X_i \sim ...
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0answers
140 views

Application of the CLT

Suppose we have $X_1,\cdots,X_n$ be a random sample with $X_i\backsim \chi^2(1)$. I'd like to show that as $n\rightarrow \infty$, $$\frac{\bar{X}-1}{\sqrt{\frac{2}{n}}}$$ is standard normal. Here's ...
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Proof of $\frac{(n-1)S^2}{\sigma^2} \backsim \chi^2_{n-1}$

It's a standard result that given $X_1,\cdots ,X_n $ random sample from $N(\mu,\sigma^2)$, the random variable $$\frac{(n-1)S^2}{\sigma^2}$$ has a chi-square distribution with $(n-1)$ degrees of ...
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1answer
231 views

Statistics Confidence Intervals

True or false: a) The center of a 95% CI for the population mean is a random variable? d) Out of 100 95% CI for the mean $\mu$, 95 will contain the mean $\mu$?
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1answer
204 views

Who invented linearization of exponential datasets to find their approximating functions?

I just learned how to find the exponential function that approximates a dataset by taking the logarithm of the data points, doing a linear regression on that data, then working out the exponential ...
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246 views

Stupid graph or stupid me?

Reading the Microsoft TechNet article "Test results: Extra-large scenario (FAST Search Server 2010 for SharePoint)", I came across this graph. Now, am I stupid (and if so, could someone please ...
3
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3answers
206 views

Using Correlation for mouse gesture recognition

I am in need to build a mouse gesture recognition system which will compare given recognition to the the gestures in training data and will say where a given gesture best fits. I am planning to use ...
2
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1answer
384 views

Poisson random variable with parameter lambda~uniform(0, 10)

Suppose $N\sim Poisson(\lambda)$ and $\lambda\sim Unif(0, 10)$ what is the expected value $E(N)$? I feel like I should just plug in the expected value of Unif(0, 10) but that seems too easy. Anyone ...
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0answers
199 views

Algebraic Geometry and Statistics

In order to do statistical inference on a model, it must be identifiable. In other words, let $\mathcal{P} = \{P_{\theta}: \theta \in \Theta \}$ be a statistical model. Then the model if identifiable ...
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1answer
2k views

Range of coefficient of Skewness

Prove that Pearson's second measure of Skewness that is Skewness = 3(Mean - Median)/Standard deviation lies between -3 and +3
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2answers
709 views

What is the relationship between the Boltzmann distribution and information theory?

I'm reading a paper on Boltzmann machines (a type of neural network in Machine Learning), and it mentions that "The Boltzmann distribution has some beautiful mathematical properties and it is ...
0
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1answer
20 views

What is the length of a sample over the length of the dataset

I need to refer to this quantity in a report, but I do not know what it is called. Basically if a dataset say: $D = \{1,2,3,4,5,6,7,8,9,10\}$ has $10$ elements in total so $|D| = 10$ and I take a ...
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1answer
341 views

Log Normal Distribution best fit

Compsci dude here. I'm trying to model the price distribution of a category of goods, and figure a Log Normal Distribution would provide a good fit. The problem is, I don't know how to do this with ...
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4answers
860 views

How to normalize a set of real numbers?

Let $S$ be a finit set of real numbers (well not exactly "real" numbers but as "real" as it is possible by representing them by the primitive type "float" on the computer). I need to find a function ...
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2answers
111 views

Finding a narrower confidence interval for a given CI, sample mean and size

I'm trying to understand confidence intervals but having some trouble. I've been doing some exercises I found online and I'm stuck on this question: I have been given a 95% confidence interval for a ...
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1answer
78 views

Game statistics: Extracting interesting patterns out of users and level

I made a small game and in course of time collected fair amount of data between users and level The level chart is long (120 levels) but looks somewhat like this $$ \begin{array}{|c|c|} ...
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2answers
312 views

Probability of market movement / Trends

If you imagine a scale from -100 to 100, if the market has moved up from 0 to 40, what is the probability is will continue to 100? There is a 50-50 chance to move up or down from 0, but what is the ...