The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.

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Doubts in Bayes' Theorem

I meet one problem on the probability and statistic theory. "Assume given the probability spaces $(X,S,\mu_i)$, $i=1,2$, and the probability space $(X,S,\lambda)$. And there exsit functions ...
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Most powerful test for discrete variable

The discrete random variable X has the following probability distributions under $H_0$ and $H_1$ $$\begin{array}{r|rrrrrrrrrr} x&1&2&3&4&5&6&7&8&9&10\\ ...
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Calculating that confidence that pairs of lightbulbs are independently illuminated.

So, you're sitting in a dark room, and on the far wall you see $n$ lightbulbs mounted above plaques numbered $1$ through $n$. There is a lightswitch on the arm of your chair. Every time you flip the ...
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67 views

Teaching Student's distribution

While it is fairly straightforward to show the basics of the normal distribution in a first year undergraduate course, how does a teacher provide good intuition when the Student distribution comes in? ...
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$E(g(X)), E(g'(X)) <\infty $ implies $\lim_{x\rightarrow \infty} f(x)g(x)= 0$ ($f$ is the density of $X$)?

I am trying to figure out the Stein's identity which asserts that for r.v $X$ having pdf $$p_\theta(x)=\exp\{ \theta T(x)-A(\theta)\}h(x)$$ where $ T$ is differentiable and $g>0$ is ...
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Finding a general form of the density function when we have a four dimentional random variable.

Consider a subject having time of the specific event $T_i$, which is a single sample from a distribution $F_i$ with density $f_i$ and support $[t_{\min},t_{\max}]$, for $i= 1,\ldots,n$. Let these ...
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27 views

Stratified Sampling: Total and mean estimate of population

Question: In a sample survey designed to estimate the total number of cattle, the universe of 2072 farms was stratified into 5 strata on the basis of the total acreage of farms. In the hth stratum (h ...
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15 views

Sampling from a graph

Suppose you have a graph $G=(V,E)$ that is unobservable globally and you wish to take a sample from the vertices of that graph to infer something about its global properties from local properties. ...
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Sum of random variables - is there an efficient way to do inference?

Suppose I have a series of variables $X_1, X_2, \ldots, X_n$. I have $S = \sum_i^n X_i$. Now suppose that I have a constraint on the distribution of S, for example from some data. Looking at any of ...
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94 views

The distribution of the ith order statistic for discrete random variables

Assume $(X_i)_{i=1,...,n}$ are a sequence of real iid random variables with continuous density $p_x$. We know that $$Y:=\sum_{i=1}^n 1\{X_i\leq u\}\sim Bin(n,F_x(u)),$$ since $1\{X_i\leq u\}\sim ...
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Normalizing multiple different features from unknown distributions

I'm doing some "exploratory" data analysis over a large set of classes/proteins, with a few hundred different features (I.E. Continuous variables) extracted from the data. The features are calculated ...
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64 views

Estimating a sub-population characteristic based on independent samples without replacement

Let a bag have 1000 balls of arbitrary colors and unknowns sizes ($r$). Suppose we also known the total volume occupied by the balls ($t_v$). We want to estimate the total volume occupied by red balls ...
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75 views

Nikolski class of probability measures - Metric and Topological Properties

I am reading a book about non-parametric statistics (Tsybakov's Introduction to Non-Parametic Estimation), and in order to prove some important inequalities on mean-squared error, different classes of ...
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118 views

hypothesis testing practice question

This is a practice question for a final exam. Two types of cordless weed-trimmer batteries are tested. One group of 5 batteries averaged $1.4$ hours, while the other group consisting of 8 batteries, ...
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55 views

degrees of freedom for a chi squared goodness-of-fit test

For a statistics project, I gave out a 20 question multiple choice quiz with each question containing five answers. I would like to run some hypothesis tests on the data by using a Chi squared ...
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96 views

Probability distribution for a digit of a number

If someone choose a digit $\alpha$ and a digit $\beta$ independently. Each one can be in $0,1, ...,9$. So $\mu = \alpha \beta$ (e.g. if $\alpha = 5$ and $\beta = 3$ then $\mu =53$). And I observe a ...
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How do I must do it?

For a sample of size n from a random variable with density function $$f_{\theta}(x)=\frac{2x}{\theta^2}, x>0$$ find the confidence interval for $\theta$ of average length consistently lower level ...
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Inference the time a process would take depending on the number of threads

I have several datasets consisting on: Number of threads n Start process time t1 Stop process time ...
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186 views

Is there a statistical hypothesis test that uses the mode?

Is there a statistical hypothesis test that considers the mode rather than the mean or median?
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Statistical sample with age ranges. How to extrapolate it using the real age distribution over population.

I have a data set consisting in the classification of the numbers of suicides by age range. I want to figure out if there is or not association between the number of suicides and the age range. But, ...
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68 views

$\mathsf kth$ moment of the standard deviation about the origin from a $\mathsf N(\mu,\sigma^2)$ population

Let T be the standard deviation of a random sample of size n from a $\mathsf N(\mu,\sigma^2)$ normal population. Find the $\mathsf kth$ moment of T about the origin, and state the condition for the ...
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Simulate from a distribution using Metropolis-Hastings and Rejection Sampling?

We have covered the basics behind rejection sampling as well as Metropolis-Hastings from class, but I am not sure how to use the two in conjunction to solve the following problem: Given $\pi(x) = ...
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How to account for both between-subjects and within-subjects covariables?

I have a data set I'm trying to analyze and I can't figure out how to include the two different kinds of covariables in my analysis. Without the covariables, the analysis isn't too complicated. It's ...
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Outlier Contained in Prediction Interval (Tme series Forecasting Problem)

In my stats class today, the professor was showing us some output from MINITAB on a prediction interval that was calculated (from time series data using standard linear regression). For one of the ...
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Uniformly Most Powerful

Let X1;X2; : : : ;X10 denote a random sample of size 10 from a population which has an exponential distribution with parameter ; > 0, i.e. with pdf f(x) =   e 􀀀x if x 0; 0 otherwise. (a) Find ...
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Show that $Y = 2\sqrt{X_1 X_2}$ has a $\Gamma(2p, 1)$ dist.

$X_1$ and $X_2$ are independent with $\Gamma(p, 1)$ and $\Gamma(p + 1, 1/2)$. Show that $Y = 2\sqrt{X_1 X_2}$ has a $\Gamma(2p, 1)$ dist.
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Sum squared errors normal

Let $X_1,..,X_n$ be independent normal random variables with common variance $\sigma^2$ and means $a+bc_i$ (where $a,b,\sigma^2 $ are constants $>0$). If $s_1,s_2$ are real numbers minimizing ...
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35 views

Comparing models to smoothed data

I am attempting to fit a model to a noisy data set. I am performing this modeling in two stages - first, smoothing it out by fitting an analytic mixture model to it, and second, fitting my final model ...
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Showing mutual contiguity

The problem: Let $P_n$ and $Q_n$ be the distribution of the mean of a sample of size $n$ from the $N(0, 1)$ and the $N(\theta_n, 1)$ distribution, respectively. Show that $P_n$ and $Q_n$ are ...
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51 views

Determining the Cramer-Rao lower bound

Let $X = (X_1,\dots,X_n)$ be a vector of iid variables from the smooth density $f(x,\theta_0), \theta_0 \in \Theta \subset \mathbb{R}$. Let $L(\theta)$ be the likelihood and $I(\theta)$ the ...
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Fitting of the Lévy triple

Given a Lévy process and its triplet $(\mu,\Sigma,\nu)$ i.e. the triplet such that for each $t\ge 0$ $ X(t) = bt + W_A(t) + \int_{|x|<1} x \tilde N (t, dx) + \int_{|x|\ge 1} x N(t,dx)$ where ...
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Iterative Mean, Covariance Algorithm Convergence

The problem is to show that the following iterations converge to the vector $\mu$ and the matrix $\Sigma$. We have data in the form of nx1 vectors $\mathbf{Q}_k$, $1 \leq k \leq N$ where ...
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60 views

UMVUE using complete and sufficient statistic

Let $X_1,X_2,...,X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. I showed that $(\bar X,S^2)$ is jointly sufficient for estimating ($\mu$,$\sigma^2$) where ...
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Draw and compare the likelihood using R

The following shows the heart rate (in beats/minute) of a person, measured throughout the day: 73, 75, 84, 76, 93, 79, 85, 80, 76, 78, 80. Assume the data are an iid sample from ...
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Bayesian inference for dependent data

Is it possible to use bayesian inference technique for data which does not follow the memoryless property? What is the likelihood function and prior in this case?
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A variant of Hoeffding's Inequallity

I'm new to concentration inequalities and I have a question related to Hoeffding's inequality. Let $X_1 ~ \dots X_n$ be a set of i.i.d random variables, s.t. $E[X_i] = \mu$, $Var[X_i] = \sigma^2$, ...
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Confidence bounds given random verification

[Edits made for clarification and brevity.] I'm working on an idea for a fault detection algorithm, and I've boiled it down (I think) to the following problem. A box contains 10 balls. The balls can ...
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Why is the marginalized inverse-Wishart distribution not equal to the inverse-gamma distribution?

Given that the inverse-gamma distribution is the one-dimensional version of the inverse-Wishart distribution, why will (philosophically speaking) an inverse-Wishart distribution that originally has ...
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29 views

Estimate distance between approximated posterior and true posterior

I'm working on a paper about using graphical models to do some prediction tasks with known observations. Since the model is complicated, finding the maximum a posteriori on the true posterior ...
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28 views

Adjusting regression for small sample bias

I have a set of data points $\{x_i\}$. These data points are grouped so that (say) $i\in\{1,2,3\}$ is group $A$, $i\in\{4,5,6,7\}$ is group $B$, etc. I would like to test the null hypothesis of no ...
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Textbooks with worked examples on identifiability and regularity

I'm looking for textbooks that would have worked examples on identifiability and regularity, something similar to a Schaum's outline. Are there any textbooks that have worked examples on these ...
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43 views

Find the maximum of an integral function with respect to another function

I'm facing this statistical data analysis problem, where I have to maximize a certain statistic in order to find the optimal filtering function. I'm a little bit out of practice with the mathematics ...
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61 views

Classification problem: admissible rule is a Bayes rule for some prior $\pi$

I have a classification problem where I want to place an observation $X$ into a population described by a pdf equal to either $f_1$ or $f_2$. Given $P_{f_i}(\frac{f_1(X)}{f_2(X)}=j)=0$ for all $j\in ...
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Comparison of two error distributions to determine “goodness of fit”

I am a physicist who is a few years out of doing his last course in statistics, so I am hoping to get some advice when comparing some data I recently generated. The context is as follows. I have two ...
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Hypotesis test: $X_i | \theta \sim Exp(\theta)$ (Likelihood Ratio Test)

Construct the Likelihood-Ratio Test to test $H_o: \theta = 0$ versus $H_1 :\theta \neq 0$ supposing that $X_1, X_2,...,X_n$ are c.i.i.d random variables such that $X_i | \theta \sim Exp(\theta)$ P.S: ...
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Can posterior distribution for a continuous variable be greater than one?

This might sound a dumb question but I am really confused about it. According to Bayes' rule we do have the following: $$p(\theta|X)=\frac{p(\theta)p(X|\theta)}{\int{p(\theta)p(X|\theta)d\theta}}$$ I ...
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Hypothesis testing with poisson distribution

At a nuclear plant great care is taken to measure the employees health.These are the number of visits made by each of the 10 employees to the doctor during a calender year. 3,6,5,7,4,2,3,5,1,4 ...
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429 views

Exponential integral approximation

I have an equation that contain exponential integral of the form: $$ \begin{equation} E_k\left(\frac{a+b ~x}{c}\right) \end{equation} $$ Where $k\geq 0$ ($k=0,1,2,...$), $a$, $b$, and $c$ are ...
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Properties of almost sure convergence

If, $\Sigma$ is the population covariance matrix and $S$ is the sample covariance matrix, $p$ is the number of variables, $\frac{p}{n} \rightarrow c$ as $n \rightarrow 0$, $\frac{1}{p}|S|_{F}^{2} ...
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Nondimensionalize an equation with logs

I have an equation with logs in it. Specifically the equation takes the form: $log(Y)=\alpha+\beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3$ Where $X_1$ through $X_3$ are variables. The trouble is that Y ...