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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Adjoint of evaluation operator: Inverse Bayesian Analysis

I'm reading "Inverse Problems - A Bayesian Perspective" by Andrew Stuart and I'm stuck with working out an application (an easier form of section 3.2): Consider a random process $u: (0,1) \to \mathbb ...
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29 views

Do i get the right MLE and 90% confidence interval of normal distribution?

I think i do right in step1 above. But i wonder whether i get the right confidence interval of mu and sigma in step2?
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22 views

Two-sample permutation tests for earthquake magnitudes before and after a damaging quake

Background and data. An earthquake of magnitude 5.17 stuck near Yountville, California in the early morning hours of September 9, 2000, injuring about 25 people and doing about $50 million damage. ...
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28 views

Confidence interval of the parameter of $\exp$ and normal distribution from MLE?

I have a sample $X_1,X_2,\ldots,X_n$ If the sample is from exponential distribution, I want to use MLE to estimate the parameter $\beta$. I know the result that ...
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20 views

Hypothesis Testing Double Proportion [closed]

Students Safe Driving has targeted seat-belt usage as a positive step to reduce accidents and injuries. Before a major campaign at one college, 44 percent of 150 drivers entering the college parking ...
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24 views

Hypothesis testing for pdfs

$X$ is an observation from a pdf $f(x)$. I want to find the UMP test of size $\alpha$ for $H_0: f(x) = \frac{1}{\pi (1+x^2)}, -\infty < x< \infty$ vs $H_1: f(x) = N(0,1), -\infty < x< ...
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41 views

Bayesian Approach: Is a die from a 3-D printer fair?

In a recent post "Fair die or not from 3-D printer"on this site @Eumel reported making a die on a 3-D printer, providing data on the faces seen in 150 rolls, and wondered about "the chances that the ...
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56 views

Can a class test scores with a bimodal distribution provide statistical evidence for cheating?

I know the normal distribution can represent many things in nature. Most items are normally distributed. I recently watched a video of a professor who claims that biomodal distributions provide ...
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Comparison of Parameter estimation using maximum likelihood and Maximum entropy.

I am not sure if the question is appropriate but I want to try my luck. One can estimate a parameter using maximum likelihood and we know it is optimal. On the other hand there are methods which uses ...
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21 views

Finding the MLE of a two parameter distribution

Suppose $X_1,...,X_n$ is a random sample from the distribution with c.d.f. $$F(x;a,b)= 1 - (a/x)^b, \hskip20pt x\geq a, a > 0, b > 0$$ Find the M.L. estimators of $a$ and $b$. I tried ...
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40 views

Conditional probability and a normal distribution

Apologies as I have never studied statistics at a high enough level to be sure that I am using some vocabulary correctly. Let's suppose I draw some $\mu_{1} \dots \mu_{n}$ from a normal distribution ...
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24 views

$X_{1},X_{2},…X_{n}$ i.i.d. ~ $N(\mu,\theta)$,calculate $E[X_{1}|\bar X]$

$X_{1},X_{2},...X_{n}$ i.i.d. ~ $N(\mu,\theta)$, I know $E(X_{1}|\bar X)$ is UMVUE, but how can I calculate $E[X_{1}|\bar X]$, should I find the joint distribution of $X_{1}\&\bar X$? I think ...
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2answers
44 views

Two-sided confidence intervals and tests

From a sample of 1751 army hospitals, estimate the mean expenses for a full time equivalent employee for all US army hospitals using a 90% confidence interval given x = ...
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23 views

U ~ $N(0,1)$, V ~ $N(0,\theta)$, W ~ $N(0,1)$ are independent random variables. $X = U+V$, $Y=V+W$, find the joint distribution of $(X,Y)$.

U ~ $N(0,1)$, V ~ $N(0,\theta)$, W ~ $N(0,1)$ are independent random variables. $X = U+V$, $Y=V+W$, find the joint distribution of $(X,Y)$. Here both $X \& Y$ are follows from $N(0,\theta+1)$, so ...
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if the variable depends on $\theta$, like $x>\theta$, indicator s.t $I_{(\theta,\infty)}{x}$ be used, what if $x>0$, should I still use indicator?

For finding the minimal sufficient statistic, if the variable depends on the unknown parameter, like $x>\theta$, we should use indicator s.t $I_{(\theta,\infty)}{x}$, but if the variable is just ...
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18 views

Most Powerful Test Neymann Pearson Lemma

If anyone can help with this problem I would greatly appreciate it. I don't really understand the neyman pearson lemma and I have no idea how to calculate this. Any steps and/or explaination would ...
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23 views

Compute the Maximum Likelihood Estimator of a pareto Distribution for statistic…

Can anyone please help compute the Maximum Likelihood Estimator for: $X_1,...X_n$ iid with $f(x; \theta) = \frac1 \theta(1+x)^{\frac{-\theta +1}\theta}$ For the statistic $T(x) = (X_1,...,X_n)$ ...
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How can I prove that $T=(\sum_{I=1}^{n}x_{i},\sum_{I=1}^{n}x_{i}^{2})$ is complete for $N(\mu, \sigma)$?

How can I prove that $T=(\sum_{I=1}^{n}x_{i},\sum_{I=1}^{n}x_{i}^{2})$ is complete for $N(\mu, \sigma)$? What I thought is try to find the joint pdf of $\sum_{I=1}^{n}x_{i} \& ...
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24 views

Calculating confident intervals for the expected daily returns

This question is related to another question I asked, but this is more problem-specific. I have a sequence of A/B currency exchanges for some days. With that data I can calculate the daily returns, ...
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99 views

What is the rule of $1.96$ for estimating confidence intervals?

I have a sequence of A/B currency exchanges for some days. With that data I can calculate the daily returns, and that's what I did. I need to calculate the confidence interval for the expected daily ...
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9 views

Statistical test when testing effect of processing on multiple responses of multiple subjects

My goal is to find whether there is a statistically significant effect of some audio processing on the localisation performance of listeners. For this purpose, say I am testing the response of N ...
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1answer
7 views

if the distribution of a positive random variable $X$ form a scale family, how can I show the distribution of $LogX$ form a location family?

if the distribution of a positive random variable $X$ form a scale family, how can I show the distribution of $LogX$ form a location family? It's obviously true, but I have no idea how to prove it, ...
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31 views

solving equation with standard normal variables

Suppose that $Z\sim N(0,1)$. Now I want to find k such that $P(Z>k)+P(5Z>k)=0.05$. I want to find this $k$ numerically, but I'm stuck in which way this can be done. Hopefully anyone can help me ...
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Covariance of estimates using method of moments.

I have a $3\times1$ vector function $f(x_i;\theta)$ where $X$ is a rv and $\theta$ is $3 \times 1$ parameter vector, such that \begin{equation} Ef(X;\theta) = {\bf 0}.\end{equation} If I have a ...
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12 views

Inference of the variance of a linear model

In Ch 7.3 of the book Elements of Statistical Learning. For a linear model fit $\hat{f}_p(x)=x^T\hat{\beta}$, where the parameter vector $\hat{\beta}$ with p ...
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44 views

What are conditions necessary to calculate a confidence interval for population mean?

if we have $1250$ values(sample), are we then able to calculate $99$% confidence interval less than $0.1$ ? how do i check if its possible ? my first thought was to try going backwards in the ...
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If $\mu = 500$ and $\sigma^{2} = 49$ , two groups selected at random, $n=32$ and $m=50$

What is the probability that the 2 groups will differ in their mean scores by more than 20 percent? I'm not sure what to do here, I recall this formula: $t=\dfrac {\overline {X}_{1}-\overline{X}_{2}} ...
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10 views

Using recursion multiple times to find a mean

I had a general question regarding a statement in statistics; it is for no assignment or examination. The setting is this, we know that for given random variables $X$ and $Y$, we can compute ...
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20 views

How to “normalize” an age-of-death probability distribution with respect to births in corresponding years?

Let's suppose that we know about the age of death of n people (this being a representative sample of a given population), all died in the same year H. We can define $P(x)$, the probability of dying ...
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14 views

How to get into exponential family form

I have troubling of whether or not a given distribution can be written as an exponential family and how would I write it. For example: (a) f(x|$\theta$)=$\dfrac{2x}{\theta^2}$ 0< x < ...
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29 views

Estimating mean of Poisson distribution when given inexact data for goodness of fit test

I have a question regarding whether a probability distribution follows the poisson distribution or not. However, I'm having trouble calculating the estimated mean of the poisson (which I need to make ...
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1answer
21 views

Find the expected value and standard variance of a jointly continuous random variable

If $X,Y$ are jointly continuous random variables with the following joint PDF: $$f_{X,Y}(x,y)=\begin{cases} \tfrac{21}{4}x^2y & : x^2< y<1 \\ 0 & : \textsf{otherwise} \end{cases}$$ How ...
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Nested Sampling Confidence interval Construction

I'm trying to understand the math underlying this business of nested sample. I'll explain my problem. My goal is to estimate the total count of $25$ sub-population. From experience, I know that in ...
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26 views

The sum of two von-Mises fisher random vectors

The Von Mises-Fisher distribution is a probability distribution on the $(n-1)$-sphere. The probability density function of this distribution is of the form $f(\bf x; ...
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Sample statistics probability bernoulli trials

Problem There are two restaurants on the campus of a university. Each can feed 120 students. We know that there are 200 students attending the university who will want to eat lunch in one of the ...
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30 views

Use standard error of mean or population distribution?

Question: Marks obtained by certain number of students are assumed to be normally distributed with mean 65 and variance 25. If three students are taken at random, what is the probability that ...
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10 views

Combining two Gaussian posterior distributions from different data to refine estimated distribution.

If we apply Bayesian inference to try and determine the distribution of a multivariate Gaussian $\textbf{x}$, and we have two predictions $$ \textbf{x}\sim N(\textbf{a}_1,\Sigma _1)~~ and ~~ ...
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21 views

Standard deviation of LS-estimator

I am trying to solve this problem: Three objects have weights $m_1$, $m_2$ and $m_3$. We measure the weight of the pairs twice and get the following result: $m_1 + m_2$ have the collective weight ...
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33 views

Show that $T$ is not a sufficient statistic

Let $X_1\: , X_2$ a random sample for $N(\theta ,1).\:$ Show that $\:T=X_1 + 2X_2$ is not a sufficient statistic. I've tried to prove it by contradiction: I assumed that $T$ is sufficient. That ...
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36 views

The maximum-likelihood estimators of $\sigma^2$

A sample of size $n$ is drawn from each of four normal populations, all of which have the same variance $\sigma^2$. The means of the four populations are $a+b+c$, $a+b-c$, $a-b+c$ and $a-b-c$. What ...
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Finding statistic and significance difference

Two new types of Indian-made cars are tested for petrol mileage. One group consisting of 36 cars averaged 14 kms. perlitre, while the group of 72 cars averaged 12.5 kms. perlitre. a. What test ...
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Testing hypothesis: Null hypothesis

I had an earlier question not satisfactorily answered and I am still smarting with ignorance. Please help with this question: I can guess that the null hypothesis is: "Method 1 better for ...
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Testing hypethesis

I have search through archive questions but none is simple enough to explain the concepts to me in a self-study situation. I am new to this statistics and I find this topic difficult to understand ...
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UMVUE for $1/ \alpha$ in Weibull distribution?

Let $X_1,X_2,X_3,X_4,X_5,\dots,X_{n-1},X_n$ be a random sample from $$f(x) = \begin{cases} \frac{\beta}{\alpha}x^{\beta-1}e^{-{x^{\beta-1}}/{\alpha}} & x>0\\ 0 & \text{otherwise} ...
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11 views

Best symmetric test

Suppose $X_{1}\sim N(0,\sigma_{1}^2)$ and $X_{2}\sim N(0,\sigma_{2}^2)$ independent. I want to find the critical region for the best symmetric test: $H_{0}: X_{1}\sim N(0,\sigma_{1}^2)$ and ...
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Intuition: Distribution Transformations, finding bounds?

Can anyone please give me a normal, intuitive definition of how we find for certain variables when performing transformation of Random Variables. For example: Say $X_1...X_n$ are iid uniform(0,a) ...
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106 views

Statistics, least square method

I am having problems with an exercise. I have some observations of the random variable $Y$: $0.17, 0.06, 1.76, 3.41, 11.68, 1.86, 1.27, 0.00, 0.04,$ and $2.10$. I know that $Y = X^2$ and that $X \sim ...
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About regression model and assumptions

I have the following general regression model $$y=E_{Y|X}[y|x]+u.....(1)$$ Where $u$ is understood as the error. In the basic model there is a common basic assumption about avoid endogeneity, i.e. ...
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testing correlation coefficient in a bivariate normal distribution

How can I show that $\dfrac{\hat{\rho } \sqrt{N-2}}{\sqrt{1-\hat{\rho}^2}}$ has a t-student distribution with $N-2$ degrees of freedom. I think I have to write it as a quotient of a normal $(0,1)$ ...
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The four assumptions on linear regression

It is clear that the four assumptions of a linear regression model are: Linearity, Independence of error, Homoscedasticity and Normality of error distribution. My question is does any of these four ...