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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Standard deviation of a rate

I am trying to measure the throughput of a network. I collected sample transfer times for 1,000 trials, and then calculated the arithmetic mean and standard deviation for the data (e.g., mean = 0.2s, ...
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140 views

Choosing set of best estimators for linear least squares

I have a measured experimental dataset which is well approximated by the sum of several basis functions in linear combinations. Linear least squares of course gives me the optimal weight for each ...
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205 views

distribution of sum of independent gaussians

what is the probability density function of $|y|^2$, where $$ y = \sum_{i=1}^n a_i x_i $$ where $x_i$ are complex gaussian random variables with zero mean and unitary variance? With only two ...
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What is the standard deviation of multiple correlated random variables subtracted from another random variable?

Wiki states that standard deviation of $X-Y$ is: $$\sigma_{x-y} = \sqrt { \sigma_x^2 + \sigma_y^2 - 2\rho\sigma_x\sigma_y }$$ I have a number (say 3) correlated random variables to be subtracted ...
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Under what circumstance will a covariance matrix be positive semi-definite rather than positive definite?

I have a covariance matrix: $\operatorname{cov}(\mathbf{X}, \mathbf{X}) = \operatorname{E}[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{X} - \operatorname{E}[\mathbf{X}])^T]$ According to ...
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105 views

Forecasting based on advises from multiple advisers with various experiences(sample sizes)

I have a signal generator, which each second generates one of the letters 'a', 'b' or 'c'. I don't know anything else about this signal generator, but I suspect that there are some patterns to it. ...
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1answer
136 views

mixture of complex gaussians

I would like to compute the following probability $$ P\left( ax \geq \sum_{i=1}^n b_i y_i \right) $$ where $a, b_i$ are constant coefficients (in my case, they are positive too) and $x, y_i$ are ...
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384 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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3answers
331 views

medians and medoids are to 2, as X and Y are to 3

What is the word for the values derived from an ordered set such that the values divide (by virtue of their positions; not by their value) the set into 3 subsets that have an equal or nearly equal ...
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1answer
63 views

Logit function and probability

I was reading a paper regarding using a logistic regression to predict an outcome of interest. In the paper the logit is given as logit(p) = log[p/1-p] = b0 + b1*x1 + .... + bn*xn => Which is fine ...
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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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216 views

Weibull Distribution Behaving Weirdly?

I'm attempting to fit a cumulative Weibull curve to some psychophysics perception data. Basically, we're testing the subject's accuracy at a number of different motion speeds, and attempting to fit ...
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66 views

Property of Bayes error on multivariate Bernoulli

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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10k views

What is CDF - Cumulative distribution function?

Could someone please explain in layman's terms what CDF is? If someone could show a real-life example where this could be useful, it would be great.
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2answers
129 views

Statistics question to apply bell curve to formula

I am trying to write a formula that would give a person a score, based on a formula. It's for a video game, and I have worked up a formula using aggregate stats like total kills divided by total ...
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137 views

Trying to find a meaningful statistical indicator to reflect user interaction with a website

A given user can interact in multiple ways with a website. Let's simplify a bit and say say a user can: Post a message Comment a message "like" something on the website via Facebook (after that we ...
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1answer
1k views

Probability of a sample mean taking various values

How do you determine the probability of a sample mean being a certain value? To make this more concrete, suppose we have a population of 1000 values. the population mean is 50 and the standard ...
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1answer
295 views

Fourier (Hankel?) transform of a discrete set of radial points (question from a chemist!)

I'm sorry because I'm not a mathematician so that my question may look a little bit messy. I have tabulated values [1] of a 3 dimensional radial function $f(r)$: ...
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1answer
677 views

Statistics with overlapping periods

I've been having a lot of discussions about finance recently in which people will point to some results using overlapping time periods and claim a high degree of statistical significance. For ...
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2answers
471 views

Probability of Median from a continuous distribution

For a sample of size $n=3$ from a continuous probability distribution, what is $P(X_{(1)}<k<X_{(2)})$ where $k$ is the median of the distribution? What is $P(X_{(1)}<k<X_{(3)})$? ...
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1answer
110 views

Expectation and random sample

Let $X_1,X_2,...,X_n$ be a random sample of size $n$ from a population $N(\mu, \sigma^2)$ with $\sigma^2=4$. Further assume that $\mu \sim N(0,1)$. Find $E[\mu|X]$ and $E[(\mu - E[\mu|X])^2 |X]$. ...
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133 views

General Definition of Likelihood Function

I am wondering if there is a way of generalizing the likelihood function of some parameters $L(\theta | \mathbf x), \theta \in \Theta$ given some data $\mathbf x$ that has been observed coming from a ...
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4answers
4k views

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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1answer
137 views

how to properly calculate difference in percent

Suppose, you have x=8 y=10 Then y is 25% more than x, if x is used as the "base" value. If y ist the base, then x is 20% less than y. So if I want to say, that ...
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587 views

Determining integral's limits when combining two random variables

Let's say we have two independent random variables, $X$ and $Y$, whose pdfs are: $f_X(x)=1,0 \le x \le 1$ $f_Y(y)=1,0 \le y \le 1$ Let $W=X+Y$. I understand that: ...
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1answer
96 views

How to calculate the possibilty of an event over time

Learned statistics before, but totally forgot. Here is the engineering problem, samples will be taken like 100 times every second, the sample value means what the target is happening (event). How ...
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2answers
575 views

How to generate and use random trees?

I have the assignment to implement a random tree classifier in MATLAB. The lecture says: ...
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2answers
391 views

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
4k 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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0answers
286 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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251 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
321 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
472 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
73 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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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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174 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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406 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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722 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
20k 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
103 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
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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121 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
792 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
241 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 ...
3
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1answer
232 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
169 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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2answers
129 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 ...