Mathematical theory of statistics, concerned with formal definitions and general results.

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Probabilistic Density Calculation

I have been struggling for several days now looking for the answer of this problem: Given the example, how do they calculate the probabilistic density in 1 and 2. I would be very grateful if someone ...
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15 views

What is a good test for comparing frequency counts between two groups?

I am looking for statistical methods used to compare frequency of observations between two groups. I have two geographical locations with data on different soil types present. for example, my data ...
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12 views

Precision and Recall for recommendation system

I am working on movielens 100K movie data for recommendation system. I divide the data into test and training and calculate the precision and recall. In testing there are more than 10K users chosen ...
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19 views

3 correlated questions: Find the roots of an equation, find the inverse of a function and find the c.d.f. of a function of a random variable

I have 3 correlated questions: Find the roots of an equation, find the inverse of a function and find the c.d.f. of a function of a random variable. The questions are in the picture. Sorry for the bad ...
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43 views

Expected value of a diagonal

If I have $E[A] = B$, where $E$ is the expected value, $A$ and $B$ are square matrices and $\text{diag}$ refers to the vector of coefficients on its diagonal. In this case, what will be the value of ...
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Randomly Generating Combinations From Variable Weights

The Question I have a list A of n unique objects. Each object Ai has a variable percentage Pi. I want to create an algorithm that generates a new list B of k objects (k < n/2 and in most cases ...
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13 views

Lognormal and radial

Is lognormal distribution a class of radial distribution? P.S Gaussian, truncated Gaussian are all classes of radial distribution.
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25 views

Lognormal distribution using normal distribution inputs

I'm simulating lognormal data in R log(Y), using the mean and standard deviation of Y. This article outlines it very well! One step in the article I'm having trouble solving how it works though. ...
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2answers
54 views

Seasonal arima forecast equation

I need to compute a seasonal arima model, and make forecasts about vehicular traffic. My idea is to compute the model with R, and use the AR and MA coefficients in another application to predict ...
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20 views

EM algorithm to impute missing value for one variable

This is from Robert Hogg's Introduction to Mathematical Statistics 6th, exercise 6.6.5. p366, It says, Suppose $X_1$, $X_2$, $X_{n1}$, are a random sample from a $N(\theta,1)$ distribution. Suppose ...
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20 views

Regression Analysis and Conditional Expectation

I have to calculate a Expectation of some expression conditioned on something. I know alpha, Beta1 and Beta2 in the below expression. Income and Family_Size are 2 Uniform Random Variables between 0 ...
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1answer
59 views

Ranking based on binomial data (example: website conversions)

Let's say I have a database of all websites and their corresponding viewers and buyers. I want to rank the websites based on the conversion rate or where we see more conversion. So example: ...
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59 views

Approximating the distribution of a linear combination of beta-distributed independent random variables

This question is related with these other two questions in Cross Validated, which has been already answered: Approximate the distribution of the sum of ind. Beta r.v Central limit theorem when the ...
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27 views

what is the distribution of $\max\{A+C,A+D,B+C,B+D\}$, given A,B,C,D are i.i.d following distribution $f(x)$?

Given $A$,$B$,$C$,$D$ are i.i.d following distribution $f(x)$, what is the distribution of $\max\{A+C,A+D,B+C,B+D\}$,? Though $A$,$B$,$C$,$D$ are i.i.d but $\{A+C,A+D,B+C,B+D\}$ are correlated. Many ...
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1answer
50 views

How to derive the recursive equation for back propagation for neural networks for $\delta_j = \frac{\partial E_n}{ \partial a_j} $

I am following the derivation for back propagation presented in Bishop's book Pattern Recognition and Machine Learning and had some confusions in following the derivation presented in section 5.3.1. ...
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24 views

The infamous cross device identification and how to go about it

I am trying to work on a cross device user identification problem wherein I have mobile data,that is apps visited and mobile attributes; as well as browser data via cookies, that is websites visited ...
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1answer
45 views

Prove that the distribution of $Q$ is chi-squared with $p_2$ degrees of freedom

Suppose $X$ is a $p$-dimensional vector following $N_p(\mu,\Sigma)$ distribution, where $\mu$ is $p$-dimensional and $\Sigma$ is $p\times p$. Let $X=\left(\begin{array}{ccc}X_1\\X_2\end{array} ...
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1answer
33 views

Coming up with a statistically accurate formula

I've just recently become fascinated by the world of statistics and I have a question. I want to come up with a formula to evaluate quarterbacks in the NFL based on 5 independent variables, which lead ...
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66 views

Finding p-value based on Poisson Distribution

I'm having trouble understanding my homework that involves p-values. Here is the context of the problem before I discuss the processes I've gone through: Suppose a company which produces fire alarms ...
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397 views

When to use Poisson v. Geometric v. Negative Binomial; GLM, the Three Count Variable Distributions

I'm trying to layout for myself when it's appropriate to use which regression type (geometric, poisson, negative binomial) with count data, within the GLM (only 3 of the 8 GLM distributions are used ...
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1answer
33 views

Questions about V(XY) , X and Y are independent

I don't really understand the part How is the formula be derived? Isn't V(S)=(E(N^2))(E((X^2))−(E(N)(E(X))^2 as X and N are independent?
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31 views

Data distribution problem

I am having the transaction dataset which contains 95% or user initiated transactions and 5% fraud transactions. When I am fitting the logistic regression model, it gives me bias prediction - as most ...
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22 views

Similarity score between athletes

I have a question about finding a similarity score between athletes. I have players that each have 6-10 variables each. How can I calculate a similarity score as a percentage in order to compare them? ...
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1answer
76 views

Conditional probability

I have been working on a question for hours now and I thought I would ask stack exchange. "Deer ticks can be carriers of either Lyme disease or human granulocytic ehrlichiosis (HGE). Based on a ...
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1answer
59 views

Chernoff Bound: Prove that $P[u \geq \alpha] \leq (e^{-s\alpha} U(s))^N$

Let $u_1, ..., u_N$ be random variables, and let $u = \frac{1}{N} \sum_{n=1}^N u_n$. If $U(s) = E_{u_n}(e^{su_n})$ (for any $n$), prove that $P[u \geq \alpha] \leq (e^{-s\alpha} U(s))^N$. $s > 0$ ...
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33 views

Relation between variance (covariance) and Fisher information when the estimator is a random variable

I have to answer a little question for my thesis work about statistics. I'm currently working with the following linear model: $$ u_j = x_j^{\mathrm{T}} \beta + \varepsilon_j,$$ where $u_j$ is the ...
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1answer
102 views

Transformation of a random vector

I am very familiar with the formula $$f_{Y}(y) = f_{X}(x(y))\left|\dfrac{\text{d}}{\text{d}y}[x(y)]\right|$$ where $X$ and $Y = g(X)$ is a one-to-one transformation of $X$. (I use $x(y)$ to mean "$x$ ...
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22 views

Estimating multiple populations

I have a set of sequential data of unknown size randomly spread across $n$ locations. I am trying to estimate the population size of all $n$ locations and provide a confidence interval as well. This ...
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29 views

How do I combine various factors/variables to a single factor/variable

I have various products and for each product I have 5 types of cost(not just monetary cost) variables associated with it, the value of each variable for any product is a positive integer. I want to ...
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70 views

Does binomial distribution have the smallest possible variance among all “reasonable” distributions that can model binary elections?

Imagine an election where $n$ people make a binary choice: they vote for A or against it. The outcome is that $m$ people vote for A, and so A's result is $p=m/n$. If I want to model these elections, ...
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26 views

How can I calculate $\int^{c}_{-c}\Phi\left(\frac{w-a}{b}\right)\phi(w)\,\mathrm dw$

This is related to someone else's question (link): Suppose $\phi(\cdot)$ and $\Phi(\cdot)$ are density function and distribution function of the standard normal distribution. How can one calculate ...
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34 views

What is a comprehensive alternative set of books that can substitute the three volume set of Kendall's books?

What is a comprehensive alternative set of books that can substitute the three volume set of Kendall's advanced statistics books? I. Distribution Theory IIA. Classical Inference & the Linear ...
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99 views

Conditional expectation everywhere non-zero while unconditional one zero?

I have a real-valued random variable $X$ that takes on positive and negative values, and $$E(X)=0 \tag{A}$$ There is also another real-valued random variable $Y$, not independent from $X$, neither ...
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50 views

How statistics changes in the course of time

I wanted to ask a question related to the choice of statistical literature. Recently I discovered Lehmann's "Testing statistical hypotheses". But! What embarrassed me was the year it was first ...
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minimum sample size to estimate chi square population mean

Let's say I have a chi square population, I required to evaluate the minimum sample size to estimate the mean with confidence level 1-alpha, I remember that Confidence level calculation based on ...
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3answers
163 views

Best loss function for very sparse real-valued data

Suppose the target output of my data prediction model is an $M\times N$ matrix where $95\%$ of the values are $0.0$ and the other values are anywhere between $0.0$ and $1.0$, what would be a good loss ...
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27 views

Question in conditional probability

Let $X$ and $Y$ be two independent random variables of discrete type with the p.m.f. $f_1(x)=\,_{n_1}\!C_x\,p^x(1-p)^{n_1-x}$, $x=0,1,...,n_1$ and $f_2(x)=\,_{n_2}\!C_y\,p^y(1-p)^{n_2-y}$, ...
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130 views

An alternative to *The Statistical Sleuth*?

I will be starting graduate school in statistics in August. I purchased The Statistical Sleuth recently [you can view the table of contents by clicking on the "Look Inside" link], and I do think it is ...
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52 views

Which Goodness of Fit Test to use when

I would like to know which goodness-of-fit test I should use and when. My choices are Anderson-Darling, Kolmogorov-Smirnov, Kuiper, Pearson $\chi^2$, and Watson $U^2$, and Cramer von Mises. My data ...
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29 views

Chi square test of independence. Where counts are coming from?

I am doing a little project applying skills learned in SAS while in school. I have two categorical variables with 7 levels in each. First variable is day of the week and the second is type of email. I ...
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1answer
28 views

Identifying a stochastic trend model

My question is a bit general Say I am given a time series $X_t$, In what ways I can use in order to check whether the sequence behaves like a stochastic trend model or not? and if yes how can I find ...
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2answers
124 views

A strange step on a proof about the distribution of quadratic forms

The following theorem comes from the 7th edition of "Introduction to Mathematical Statistics" by Hogg, Craig and Mckean and it concerns the necessary and sufficient condition for the independence of ...
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51 views

What statistical test should I use between two scenarios?

Scenario: I have a test data of response time of two websites in two different servers. Let's say website A and website B are tested on server A and server B not respectively but website A and website ...
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57 views

Distribution of $X'X$ if $X\in\mathbb{R}^{T \times N}$ and $X_i'\sim N(\mu,\sigma^2I_N)$

Let $x_i\in\mathbb{R}^N$ be multivariate normal distributed with mean vector $\mu\in\mathbb{R}^N$ and no correlation: $x_i\sim N(\mu,\sigma^2 I_N)$. Given $T$ iid samples, define the matrix ...
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11 views

Gap test for random number generators

Gap test is discussed here http://math.stackexchange.com/questions/270377/gap-test-for-random-numbers/1301443?noredirect=1#comment2643072_1301443 , if we have probabilities how we can find the ...
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1answer
35 views

Why pnorm function and sum of dnorm function differs for same input?

Data is height of men in 'inches'. Mean is 70. SD is 3.3 I wanted to compute the probability of men whose height is in between 69 inches and 74 inches. The following methods gave me different ...
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3answers
72 views

Understanding Sampling with replacement

I am doing Stat110 and in the book, "Introduction to Probability" they give the following definition of sampling with replacement: Theorem 1.4.5 (Sampling with replacement). Consider n objects and ...
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37 views

How to design an economic index?

I'm designing a linear economic index: $I_{t}=W_{1} X_{t_{1}}+W_{2}X_{t_{2}}+W_{3}X_{t_{3}}$ where $W_{1}$ to $W_{3}$ are weights of inputs ($X_{t_{1}}$ to $X_{t_{3}}$) and $I_{t}$ is the final ...
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1answer
74 views

Most accepted theory to analyse trend in data series

I have pollutant concentration for long period I want to determine trend. I have read some of the question answers in this blog regarding same . I have few queries. What is most accepted theory to ...
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35 views

Multinomial distribution pdf and fisher information matrix

The question is from Robert Hogg etal's Introduction to Mathematical Statistics 6th Edition Page 347 Example 6.4.5. It says: Consider a random trial which can result in one, and only one, of k ...