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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461 views

Transitivity of uncorrelated random variables?

Suppose $cov(X,Y)=0\;$ and $\;cov(Y,M)=0$. Does this imply $cov(X,M)=0\;$, if all distinct RV are normal?
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159 views

Is There a $R^2 \rightarrow R^2$ Linear Transformation to Make XOR Problem Separable?

By Cover's Theorem(1965), it is possible to make patterns separable if the original feature space is transformed to a higher-dimensional space. Think of the XOR problem. It is not possible to ...
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153 views

Average Maximum Of Random Subset?

Suppose I have a set of $n$ real numbers, $\{x_1, x_2, \dots, x_n\}$. I choose a uniformly random subset of size $m \le n$. What is the expected maximum of the subset in terms of $n$, $m$ and $x_i$? ...
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1answer
254 views

Is Fuzzy Logic Needed? [closed]

I had a very big doubt in my mind about Fuzzyness. When statistics is answering all the questions, which we see generally in Fuzzy theory. Then why one SHOULD learn Fuzzy Theory. Or is there any gap ...
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40 views

Confusion regarding ML estimate

I was going through this article and they have this log likelihood given by $$ LL = \sum_{i=1}^n A_i\log p_i + \sum_{i=1}^n A'_i\log(1-p_i).$$ Basically this is the loglikelihood of a logistic ...
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343 views

How should I calculate “fill rate”? People inquire about an item and then buy it days/months later?

So I have posts(people posting a want for an item) and then I have fills(the person actually getting the item) There are hundreds of posts per day but the fill for that item can be days or weeks away ...
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1k views

Maximum Likelihood Estimation of an Ornstein-Uhlenbeck process

I am wondering whether an analytical expression of the maximum likelihood estimates of an Ornstein-Uhlenbeck process is available. The setup is the following: Consider a one-dimensional ...
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57 views

Taking averages

Suppose I have a function $f(\theta)$ that is a function of the angle $\theta\in [0,2\pi)$. Why is the average of $f$ over a large collection of randomly oriented objects: $$\int f(\theta)\sin ...
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197 views

Unimodal marginals imply the following on the joint distribution?

Suppose $X_1,\ldots,X_n$ are each unimodal random variables with mode around zero. In particular, for each $i$, $X_i=X_i'-X'$ where $X'_i$ and $X'$ are i.i.d. unimodal random ($X_i'$ and $X_j'$ are ...
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1answer
167 views

Probability distribution explanation

What exactly is a probability distribution, and what are the two requirements for a probability distribution? I am not sure what this means or how to apply it? Any examples that can be given would ...
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108 views

What does $(\text{Second Moment})^3+(\text{Third Moment})^2$ equal?

I have an equation as follows \begin{equation} \begin{alignedat}{1} \left(-\frac{\mu_2(\alpha)}{2}\right)^3+ \left(\frac{\mu_3(\alpha)}{2}\right)^2 \end{alignedat} \end{equation} where ...
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64 views

Finding an accurate rating between 1 and 10 based on numerous data points

I'm going to provide a fictitious setup that aligns with my mathematical needs. I am a company that is measuring the bounciness of balls and providing two 1-10 rating to each ball based on how it ...
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1answer
435 views

Statistics: How to measure how accurately probabilities are reported?

If you roll a six sided die a bunch of times, and count how many times the number 1 shows up, you'd expect it to show up about 1/6 of the time. Now if you roll this die 1000 times, and the number 1 ...
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2answers
883 views

A question about a proof of Neyman's factorization theorem

This question comes from the proof of Neyman's factorization theorem in Robert V. Hogg, Joseph W. McKean, Allen T. Craig, "Introduction to Mathematical Statistics", 6th edition, pp 376-377. In the ...
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1answer
64 views

Types and Typical sequences

Joint types can often be given in terms of the type of x and a stochastic matrix \begin{equation} V:X\rightarrow Y \end{equation}such that $ P_{x,y}(a,b)=P_{x}(a)V(b|a)$ for every $a\in X$ , $b\in Y$. ...
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1answer
107 views

analysing time series

I'm analysing time series and have a question related to the dependency between the elements. Lets assume I have a time series and want to extrapolate future values. For this purpose I want to know if ...
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1answer
4k views

Quantile and percentile terminology

Note: This is answered by user974514 below, but there was some discussion outside of the "answer", so I paraphrased the final answers inline here. I've asked around for the exact usages of the terms ...
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1answer
156 views

Estimating missing values in a dataset and averaging values

I have a statistical maths problem where I have a rather large dataset consisting of a timestamp (rows) and a quantity of - lets say detections - per each day (columns for each location). Currently I ...
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1answer
796 views

approximation hypergeometric distribution with binomial

Let $X$ be $\rm{Hypergeometric}(2n,\ell,n)$ and $E(X)=\frac{1}{2} \ell=:\mu$. Is it possible and how to approximate the $q$-th central moment $E(X-\mu)^q$ of the hypergeometric distribution by the ...
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10k views

comparing distribution of two data sets

I need to compare the distribution (unknown) of a set of data to the distribution of another one (unknown). In particular, I want to check for equality of the two distributions. What are some ...
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1answer
1k views

What is the probability of two people meeting?

I am trying to figure out a solution to the following problem: Let there be two groups of people, Group A and Group B. Group A represents x percent (e.g. 1%) of the world's population, and Group B ...
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70 views

how to weigh data according to given distribution

Here is what im trying to do: Given $n$ arbitrary data points $(x_i, y_i)$ and a distribution, say $N(0,1)$. I need to produce output $(x_i, y^*_i)$ such that $y^*_i$ are weighted versions of $y_i$, ...
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2answers
401 views

“Fully correlated” definition

Really sorry to be a noob, but I'm a programmer, not a mathematician, and all of my knowledge about statistics come from this book "Schaum's Outline of Theory and Problems of Probability, Random ...
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283 views

A convex programming problem involving sum of logarithms of linear functions

Here is a convex programming problem I encountered while working on an estimation problem for a mixture of multinomial distributions. We have a matrix $A_{m \times n}$ containing non-negative real ...
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2answers
151 views

Proving that the magnitude of the sample correlation coefficient is at most $1$

How can you show that the magnitude of the sample correlation coefficient is at most $1$? The formula is huge, I'm not even sure how to approach this. Can anyone point me in the right direction? ...
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1answer
93 views

Expression for $n$-th moment

I stumbled upon an expression in an article of statistics for an $n$-th moment with $X$ being a random variable over $[0, \infty)$. $$\mathbb{E} X^{n} = \int^{\infty}_{0} nz^{n-1}\; \text{Pr}(X > ...
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65 views

What are the chances of having certain cards on certain turns if you draw one card each turn?

You have a sixty-card deck. It has: 3 Aa and 3 Ab cards (6 A cards) 4 B cards 4 C cards 4 Da and 2 Db cards (6 D cards) 14 Ea and 10 Eb cards (24 E cards) At the begining of the game, you draw 7 ...
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1answer
66 views

How to compute sample variance from sample moments

Given $X_1, \dots , X_n$ i.i.d. and the two sample moments $$M_1 = \frac{1}{n} \sum_{i = 1}^{n} X_i = \bar{X}$$ and $$ M_2 = \frac{1}{n} \sum_{i = 1}^{n} X_i^2$$ how can I compute: $$ S^2 = ...
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78 views

Deducing that the sum of exonentials has a gamma distribution

If I have got $\ X_1,...X_n $ all independent and identically distributed with the exponential distribution with parameter $\ 1/\theta$. Then how can I deduce that $\ S=X_1+...+X_n $ has the gamma ...
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1answer
90 views

is the approximation of the sum true?

Someone commented under my question Calculation of the moments using Hypergeometric distribution that $$ \sum_{k=0}^l\frac{{l \choose k}{2n-l \choose n-k}(2k-l)^q}{{2n\choose n}}\sim \sum_{k=0}^l ...
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2answers
129 views

Calculating statistic for multiple runs

I have s imple, general question regarding calculating statistic for N runs of the same experiment. Suppose I would like to calculate mean of values returned by some Test. Each run of the test ...
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1answer
322 views

Density function $Y= \max(X_1, X_2)$

If $X_1$ and $X_2$ are independent random variables each of which has density function of the form: $$f(x)= \Bigg\{ \begin{array}{cc} 2x;&0<x<1\\ 0; & \text{otherwise} \end{array} $$ ...
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1answer
182 views

linear kernel pca get corresponding dimension

I am implementing my own version of linear kernel principal component analysis for better understanding the algorithm. I faced a problem which seems to be specific ...
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1answer
158 views

Mathematical idea behind tax bracket

This may be a lame question. The tax bracket is a way to calculate the tax based on the taxable income. For example Imagine that there are three tax brackets: 10%, 20%, and 30%. The 10% rate ...
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2answers
861 views

Examples of Student's T-distribution in real world empirical data?

I have recently stumbled onto some empirical (forecasting error) data that should be normally distributed. However, the normal distribution fits relatively poorly due to the abundance of data points ...
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1answer
372 views

Multiple Regression over an experimental dataset

I want to do a multiple regression over an experimental result shown as 3D-Plot and heatmap in following Images. Sorry as a new user i am not allowed to post them directly but it is just a link to ...
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2answers
145 views

Compute the probability that $ | \bar{X} - \mu | > S$

Given $X_1, \ldots, X_n$ from $\mathcal{N} (\mu, \sigma^2)$. I have to compute the probability: $$P\left(|\bar{X} - \mu| > S\right)$$ where $\bar{X}$ is the sample mean and $S^2$ is the sample ...
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1answer
160 views

Measure-theoretic view of expectation of a Bernoulli sequence

Problem: I have a good understanding of basic Bernoulli and Binomial RVs, but this was foundational work in statistics. I am attempting to try and apply my (minimal but increasing) knowledge of ...
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1answer
33 views

For System dependent on normally distributed parameter, are deviations added or variations?

Say, A and B are two normally distributed parameters with their variations being $\sigma^2_a$ and $\sigma^2_b$. Now for system C, which is linearly dependent on these parameters, is its ...
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1answer
2k views

Distribution of the sample mean of a exponential

I please ask someone to check if my calculations are right. I have $X_1, ..., X_n$ from a $\mathcal{E}(\lambda): f(x, \lambda) = \lambda e^{-\lambda x}$. I have to find the $k$ such that $P(\bar{X} ...
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1answer
70 views

Variation Tolerance

I came across a statement in my course book that 3$\sigma$ is considered as a means of tolerance. Can anyone explain it to me. I understand that +3$\sigma$ to -3$\sigma$ constitutes 99% of the ...
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1answer
91 views

Touch Typing Index - Speed and Accuracy

I am trying to determine the ability of my students to touch type. I have data on their speed (in seconds) and their accuracy (number of errors). I also know the number of words in the test (50 ...
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0answers
445 views

Calculating a fisher information

$\ (X_1,X_2,X_3,X_4)$ has a multinomial distribution with parameters 3$\theta$/5, $\theta$/5,$\theta$/5,(1-$\theta$) Calculate the fisher information where $\theta$ is in [0,1]. So I wrote out the ...
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1answer
103 views

Linear regression for normal distributions

Basically, I have that $\ Y_i = \alpha +\beta(x_i-x_{bar}) + \epsilon_i $ where $\epsilon_i$ are i.i.d normally distributed with mean 0 variance $\sigma^2$ $\ Y_i ~~has ~a~normal~distribution~as ...
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1answer
412 views

Fitting of Closed Curve in the Polar Coordinate.

I know how to fit a curve when given some data points in the cartesian coordinate. Recently, I encountered a model that needs to fit a closed curve in the polar coordinate. I'm thinking of deducing a ...
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34 views

Dynamic Light Scattering

In DLS does the combination of the Time Average and Ensemble Average give a better statistical average than the results shown by each case considered separately ?
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983 views

A question on confidence

So, I've been reviewing some of my old stats courses in preparation for an interview I have in a couple of days. I'm a bit stuck on a particular question and hope you could help. A drug trial gives ...
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0answers
63 views

calculation of the sum using idea of one answer

I am wondering if the sum (the $q$-th moment) in my question Calculation of the moments using Hypergeometric distribution can be calculated using idea in Evaluating 'combinatorial' sum ? ...
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1answer
613 views

Calculation of the moments using Hypergeometric distribution

Let vector $a\in 2n $ is such that first $l$ of its coordinates are $1$ and the rest are $0$ ($a=(1,\ldots, 1,0, \ldots, 0)$). Let $\pi$ be $k$-th permutation of set $\{1, \ldots, 2n\}$. Define ...
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5answers
639 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 $D$, ...