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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Intuitive explanation of requirement for achieving the Cramer Rao Lower Bound

this question relates to the requirement for achieving CRLB. I know that for a random sample $Y_1, \ldots, Y_n$, an estimator $U$ of $g(\theta)$ is MVUE (i.e. it is unbiased and also ...
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40 views

calculating $E(X_{(i)}| \sum_{i=1}^5 X_i)$

suppose 5.5,3.5, 2.5,4.5,2 be a random sample from of gamma distribution with parameters of $ \beta,\alpha=2$. if $Z_{(i)}$ be i-th order statistic a random sample of size 5 from $\Gamma(2,1)$, how ...
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38 views

What is the appropriate statistical test to see if a quantity has been distributed differently into discrete bins?

Say I have $10^6$ balls, $3$ bins $A,B,C$, and $2$ machines $X$ and $Y$ that distribute the balls into the bins according to an internal set of rules (i.e. a probability distribution). If I run both ...
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39 views

Function Looks Poisson-Like: But What's the Parameter $\lambda$?

(On pause) I have $$f\left(x\right)=-x\left( x\sqrt{4-x^2}-4\arccos\left(\frac{x}{2}\right) \right)\arccos\left(\frac{x^2+d^2-1}{2dx}\right)$$ which looks a bit like the continuous version of ...
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66 views

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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114 views

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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63 views

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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71 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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29 views

How sample size affects confidence interval.

Suppose the weight of n primary one students has sample mean of 20KG. If n = 40, a certain percentage of confidence interval for the population mean is (15.5,24.5). Find the confidence interval if we ...
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43 views

t-distribution and Degrees of freedom

Why t- distribution have n-1 degrees of freedom? I know that it is used when population variance is not known but what determines n-1
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41 views

Uniform Boundedness: Am I right or my TA?

I am a student, and I disagree with the solutions our TA has prepared. I am seeking verification that I am correct or explanation as to why I am wrong. It seems to be a disagreement or ...
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36 views

$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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26 views

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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41 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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22 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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23 views

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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143 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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42 views

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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68 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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95 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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135 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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60 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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99 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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68 views

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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28 views

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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235 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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17 views

How can we have $T_n \xrightarrow{\mathbb P_\vartheta} \vartheta$ if $T_n$ are defined on different spaces?

Here is how I understand the standard parametric model in statistical inference: We have a r.v. $X:\Omega \to \Psi$ which has some known to us distribution yet the exact parameter is unknown to us. ...
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21 views

Conditional Probability/Expectation in the EM algorithm

I'm doing a study in which I measure data under a random censoring process. The observed data which may be interpreted as the lifetime of a subject, is denoted by $t$, with the censoring variable $c$ ...
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16 views

Fisher Expected Information for a Gaussian Process model

Suppose I have a two dimensional Gaussian process model (GP), defined by a squared exponential correlation function s.t: $$R(x_{i},x_{j}) = \exp\left(-\frac{|x_{i} - x_{j}|^2}{2}\right).$$ I am ...
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41 views

Limiting Distributions and the Weak Law of Large Numbers

I have that $Y_1, Y_2, ..., Y_n$ are i.i.d. Poisson random variables with mean 1, and that $U_n = \sqrt{\frac{\sum_{i=1}^{n}{Y_i^2}}{n}}$. Given that I have a sequence $U_1, U_2, ..., U_n$, I'm ...
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Do higher order sample moments converge to the distributional mean?

The Methods of moments estimation is based on the law of large numbers, which says that the sample means of i.i.d. random variables from any distribution converge to the distributional mean as the ...
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36 views

Hazard function for proportional odds model

The Cox proportional hazards model for survival data with covariate ${\bf z}$ is defined through the hazard function $h(t,{\bf z})$ by $$ h(t,{\bf z}) = h_0(t)~\cdot\theta~~,~~~\theta = \theta(\beta, ...
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19 views

Questions on the theory of Lasso

The linear model ${\bf Y}={X\beta}+\epsilon$, where ${\bf Y}$ is a $n\times 1$ vector, and ${\bf X}$ is $n\times p$ matrix. $n\lt p$ and $rank({\bf X})=n$. $\epsilon\sim N(0, \sigma^2)$. How to prove ...
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24 views

inferential statistics methods for population?

Is that true to use inferential statistics methods when we study whole population? I mean for example is that true to use hypothesis test when whole population are under study? Suppose I am studying ...
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42 views

$Z$ is Normal$(\sigma,1)$, find UMVUE of $P(Z\leq 0)$.

Given i.i.d. samples $X_1,...,X_n$ from Normal$(\sigma,1)$, find the UMVUE of $g(\sigma)=P_\sigma(Z\leq 0)$. I tried to use Lehman-Scheffe theorem. We now that $\sum_1^nX_i$ is sufficient and ...
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18 views

How can I tell whether sample size is inadequate or not ?

I am given sample size of 15322 students and our research topic is to find out a relationship between students academic performance and participation in sports team. The question asks " do you think ...
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17 views

Why not use always a binomial exact test to compare two proportions instead of chi square?

I am trying to figure out what test I should use in the following scenario: I know that there is a lot of room for improvement in a specific area at work - being extremely critical, let's say that ...
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16 views

how to find relationships in dataset with multiple variables

I have a large project data set ,which includes numeric values like dollar amounts, and non numeric quantities like country codes, purpose codes etc I want to find relationships between the variables. ...
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12 views

Showing that moment estimates are asymptotically bi-variate normal.

Let $X_1,\dots,X_n$ be iid $\Gamma(p,1/\lambda)$ with density $g_\theta (x) = \frac{1}{\Gamma(p)} \lambda^p x^{p-1} e^{-\lambda x}$, $x>0$, $\theta = (p,\lambda)$, $p > 0$, $\lambda > 0$. ...
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22 views

Uniformly minimum variance unbiased estimator

How to prove $ \overline{X}=\frac{1}{n}\sum_{i=1}^nX_i$ is the uniformly minimum variance unbiased estimator of $\mu$ when $X_i\sim N(\mu,\sigma^2),$ and $\sigma$ is known. Idea: Let ...
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5 views

How to parameterize some emprical data

I would like to describe a bunch of data that I have collected as a function of two variables. The data is phytoplankton absorption in my local area that has changed in concentration. The data looks ...
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44 views

$E\bigl(\frac{2}{1+x}\bigr)$ for Beta(2,$\frac{1}{2}$) random variable

Let x ~ Beta (2,$\frac{1}{2}$). Then calculate $E\left(\frac{2}{1+x}\right)$. So, ${E}[g(X)] = \displaystyle \int_{-\infty}^\infty g(x) f(x)\, \mathrm{d}x$ . $\displaystyle f(x;\alpha,\beta) ...
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How to represent data graphically in which a respondents can choose more than one group?

I am analyzing a survey in which one of the questions was (What is the industry segment of your company?) and the respondents can choose more than one category. I have used a pie chart to represent ...
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183 views

Improvement of Minimum description length (MDL) estimate.

I earnestly request apology if this question is inappropriate for the forum. The question has two parts one technical and the other is not technical. I would appreciate any response. Let me consider ...
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34 views

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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78 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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29 views

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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25 views

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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37 views

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 ...