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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Probability density function transformation

Probability density function of f is given as a uniform distribution, f(x)=1 and I need to find the probability distribution function of Y=X-X^2. What I have done so far is that I found the inverse ...
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28 views

Compute joint density function of exponential fuction

Consider a set of continuous random variablces $Y_1 ... Y_n$, i.i.d, exponentially distributed . with rate parameter $\lambda$. I showed first that for one single variablce (ie the first) its ...
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42 views

Discrete mathematics vs. non parametric statistics

Is there any meaningful connection betveen non parametric statistics and discrete mathematics? I am reading this book: ...
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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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28 views

how to prove independence in this case

The question is : $X_1,...,X_n$ are i.i.d.$Uniform(0,\theta)$. Let $X_{(n)}$ denote the maximum of these $n$ random variables. Prove that $\frac{X_1}{X_{(n)}}$ and $X_{(n)}$ are independent. What I ...
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44 views

Joint Probability of Random Variables

Suppose I took measurements $\{X_i\}$, which are all independent and they follow a normal distribution $X_i\sim N(\mu,\sigma)$. I am asked for the joint probability of all of the measurements. Based ...
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Exchangeable/Independent Bernoulli Distribution

Let P be a uniform random variable on the interval $(0,1)$ with density function f(p) = 1, $0<p<1$. Let $X_i|P$, i = 1,2,...,n be independent and identically distributed random variables having ...
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33 views

Maximum Likelihood Question

The aim is to find the maximum likelihood estimator for theta. $f(x)$ is given and we can assume that $1\le x\le-1$. I have completed the steps seen in the image, however I am having difficulty ...
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Maximum Likelihood Estimation Question

I'm really struggling with this question. From my understanding in order to find the maximum likelihood estimator for theta, the function needs to be partially differentiated with respect to theta ...
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Central Limit Theorem & Delta method problem

Let $U_1$,...,$U_n$ be a random sample from the U(0,1) a. Let $X$=-log($U$). Find the distribution of X b. Let $Y$=$1/{\prod_{i=1}^n U_i^{1/n}}$, where $U_1$,...,$U_n$ be a random sample from the ...
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58 views

Show expectation is infinite

Let $X_1,\ldots,X_n$ be independent, identically distributed with expectation 1 and finite variance. Find the limit distribution of $\sqrt{n}(\bar{X}_n^{-1}-1)$. If the random variables are sampled ...
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Qual problem: One dim. sufficient statistic for $\lambda$ based on the data (X,Y) where X-Piosson$(\Lambda)$ and Y-Bernolli$(\lambda/(1+\lambda))$

Qual problem: We observe the pair $(X,Y)$ where $X$-Poisson$(\lambda)$ and $Y$-Bernoulli$(\lambda/(1+\lambda)),$ $\lambda$ is unknown. Find one dimensional sufficient statistic for $\lambda$ based ...
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Difficult Survey Sampling question

Question: A Secretary of State wants to survey the primary owners of motorcycles registered in the state to estimate the proportion who want the license plates redesigned. (Primary owner means that ...
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16 views

Show that the found value is the MLE

Let $ X_1, ... X_n$ i.i.d with pdf $$f(x;\theta)=\frac{x+1}{\theta(\theta+1)}\exp(-x/\theta), x>0, \theta >0$$ It is asked to find the MLE estimator for $\theta.$ The likelihood function is ...
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83 views

Poisson random variables and Binomial Theorem

I'm working on a problem from Casella and Berger's Statistical Inference. X is distributed as Poisson$(\theta)$ and Y is distributed as Poisson$(\lambda)$, with X and Y being independent. We let U = X ...
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27 views

Showing something converges, in distribution, to a normal distribution

I'm not sure how relevant the first few parts are, but I will post it just in case... $(X_i,Y_i), i=1,\dots,n$ are independent where $X_i$ has an exponential distribution $\mathcal{E}(\lambda_i)$ ...
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Finding a sufficient statistic for an iid sample of the Gumbel distribution

$G(x;\alpha, \beta) = \exp\{-\beta e^{-\alpha x}\}$ for $x \in \mathbb{R}$ is a distribution (Gumbel family). Side question: is $G(x;\alpha, \beta)$ a member of the exponential family? I do not think ...
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Question about Lagrangian Multiplier (Gradient) Statistic of constrained GMM

I am trying to derive the Lagrangian multiplier statistic (GMM version) under a restriction. The question is given below The quadratic form is given by $Q_n(\theta,\alpha)=[m(\theta)', ...
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37 views

MAP for exponential function (Maximum a posteriori)

I am trying to find the MAP for an exponential function of the form $p(y) = \theta.e^{{-\theta}y}$ Given that $\theta$ is constant, I want to estimate maximum $y$ = $p(y).p(X=x_i|y)$ for $i = 1..n$. ...
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Prove that the the variance estimator $\widehat{\sigma}^2=MSE/(n-2)$ is biased is the simple linear regression model

This is in scope of the simple linear model. Im trying to prove that $\mathbb{E}\left(\widehat{\sigma}^2\right) = \sigma^2$ for $$\widehat{\sigma}^2 = \frac{1}{n-2}\sum^n_{i=1} ...
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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

Examining the effect of a quantitative factor on response.

To examine the effect of a quantitative factor temperature on yield,the researcher has a plan to use the following model for the analysis: $$y_{ix}=\beta_0+\beta_1 x+\epsilon_{ix}$$ where $y_{ix}$ ...
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15 views

Testing statistic $\frac{MSS(X)}{MSS(Y)}$

Suppose a test statistic $\frac{MSS(X)}{MSS(Y)}$, where $MSS$ denotes Mean Sum of Squares, is to be used for testing the significance of the factor $X$. Do we need the assumption $$\mathbb ...
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How one derives significance test for pearsons correlation coefficient?

I am exploring statistics and probability. What upsets me, only ready to use algorithms are present in the books. But no example how one derives a significance test, where from the test statistics ...
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25 views

Expected Residual lifetime

I have a 2 part question. I was able to figure out part 1. I need some help with part 2. I will write out part 1 (and my solution) for completion. Let $T$ be a continuous survival time with survival ...
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Determine sample size according to some unknown distribution with given error rate and confidence

Assume $x\in\mathbb{N}$ obey some unknown distribution, and I can sequentially and independently acquire infinite samples of $x$. Now, given an error rate $\epsilon$ and confidence $1-\delta$, can I ...
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21 views

Order of growth in uniform distribution

Consider an i.i.d. sample $\{X_1, \ldots , X_n\}$ from the uniform distribution on $[ 0,\theta]$ and the estimator $$M_n = \max\{X_1,X_2,\ldots,X_n\} $$ What does the above statement mean? I ...
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37 views

confidence level interpretation

A new treatment for strokes is put on trial. There are two equal size groups,one group is given a drug, and one a placebo. The 95% confidence interval for the difference between the two proportion of ...
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135 views

How to calculate peakiness or uniformity in histogram?

I have a histogram with 20 bins ranging from -1 to 1 with an interval of 0.1. I would like to know if the histogram distribution is uniform or is peaked. I want to compare several such histograms and ...
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95 views

Finding test of critical region for sum/variance of normal distributions

Let $Y_1,....,Y_n$ denote independent, identically distributed random variables such that $Y_1$ has a normal distribution with mean $\theta$ and standard deviations $\theta$, where $\theta$ > 0. ...
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Showing distribution has a $\chi^2$ distribution with df = n

Let $X_1,X_2,....,X_n$ denote independent identically distributed random variables such that $X_1$ has density $p_1(x;\theta)$ where $\hspace{15mm}p(x;\theta) ...
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statistics basic question on covariance

anyone would help me in a basic example? a fair coin is tossed, n times. X is the number of Head and Y is the number of Tails. what is the COV(X,Y).
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Statistics and Some Information Challenge

relation between two attribute x,y is $y=\alpha\beta^{-x}$. According to 8 experiments these information were gained. what is the estimation of ( $\alpha, \beta$) using Least Square Error? it's 2010 ...
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Conditional PDF Inference

I am attempting to create an inference model, such that given any $y$, I can output an estimated probability density function of $x$. Given $X,Y$ where $f_X$ and $f_Y$ are probability density ...
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Errors and Residual

Why are errors independent but residuals dependent? As far i know the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent. But also ...
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Risk function of binomial random variable

Suppose $X\sim Binomial(100,\theta)$, True estimator($\delta$) $= X/100$; $$R(\theta ,\delta) = E_{\theta}\left[\left(\theta - \frac{X}{100}\right)^2\right] = \theta(1-\theta)/100$$ I am ...
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3answers
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Variance of the sum of sample means

Let $X$ be a random variable with normal distribution with mean $ \theta$ and variance $ a>0$. Let $ Y $ be a random, variable with normal distribution with mean $\theta$ and variance $b>0$. ...
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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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299 views

Derivation of standard error of mean

I was going through this wikipedia article on standard error. I could not understand the crucial step here. It goes like this: This formula may be derived from what we know about the variance of a ...
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Proof that data-processing selection process is statistically independent of the sample?

I hope to replace a proof of my own, in a paper explaining that the "no free lunch" theorems for optimization actually address sampling and statistics, with a reference to an existing result on ...
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Does the parameter change during data generation in Bayesian Inference?

Let's assume that we have the following graphical model: This graph encodes the joint distribution $P(p,x_1,x_2,x_3,x_4) = P(p)\prod_{i=1}^{4}P(x_i|p)$. In the Bayesian inference, if we know ...
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normality of data

Does the qqplot below suggest that the data is normally distributed? The fact that it's nearly perfectly linear is to me an indication of normality. However, the Anderson-Darling test for some reason ...
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factor models and using cross sectional regression?

I have been doing some reading on factor models. In the literature it mentions that when creating a portfolio that maximises particular attributes it may lead to unwanted bias to other factors. I ...
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38 views

Finding the sample size

My main question is this: Assume I want to conduct a plebiscite of an entire town about whether or not the citizens want to have their water fluoridated or not, I want to conduct a survey first. If ...
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$Y_{(n)} = X_{(n)}/\mu$?

If $ X_1, ...,X_n$ are iid random variables such that $ X_i \sim U(0, \mu)$, is that true that if $Y_i = X_i/\mu$, then $Y_{(n)} = X_{(n)}/\mu?$ I am sorry if the question looks so simple and I am nt ...
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Show that statistic is (not) sufficient

I need to verify ifthe statistic $|X|$ is or npt sufficient for $\mu$, if $ X \sim N(\mu, 1)$ Using the definition, I've obtained the pdf of X given $ T(X)=|X|:$ $$f_{X|T}(x|t) = ...
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Reconstructing message from clippings

It's a question arising from genetics. Suppose we have a message - a list of 100 bits. Now we run a process which cuts off either one or a pair of bits (let's say with equal probability). As a result ...
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Unbiased asymptotic variance

Problem: Let $X_1,...,X_n$ be indep. r.v.'s that satisfy, for $i = 1,...,n$, $E(X_i) = \mu_i(\theta)$ & $\mathrm{Var}(X_i)= \sigma_i^2(\theta)$. $\theta$ is the parameter of interest and the ...
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55 views

a complete sufficient statistic in geometric distribution

Exponential family has a very good property that could be used to conclude if a statistic is complete: $X_1,X_2,\ldots,X_n$ are from exponential family which has the form as: $$f(x\mid \theta ...
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52 views

How to prove a statistic is an ancillary statistic?

Though with the knowledge that an ancillary statistic is a statistic has distribution that is independent of the parameter, I feel like I still don't know very well for verifying a statistic is an ...