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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How can I determine the best relationship for 3 variables, given several data points?

What is the best way to determine the relationship for three apparently related variables? The relationship does not appear to be linear, and may follow a combination of non-linear functions. I have ...
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9 views

Bayesian Inferences: Finding Posterior HPD Interval

I am currently working with Beta-Bernoulli and Beta-Binomial models. I have been searching around for the specific steps in obtaining the Posterior HPD intervals for both. Does anyone know how to find ...
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42 views

Finding the Cramer-Rao Lower Bound

Given the probability density function $$f(x; \theta) = \frac{ \left(\ln(\theta)\right)^{x}}{\theta x!}, \quad x = 0,1,\ldots ; \theta > 1$$ and $0$ otherwise, find the Cramer-Rao Lower Bound ...
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22 views

Transformation of continuous random variables

Let's say that $Y = \log T = \alpha + \sigma W$. I know that If $W$ has logsitic distribution, the $T$ will have the log-logistic distribution. Also, if $W$ has the standard normal distribution then ...
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24 views

If the conditional expectation of the random variable does follow a linear function, can we show the probability a particular data set happens?

Suppose that $\mathbb{E}[Y\mid X=x]=\beta_0+\beta_1x$ where $X, Y$ are random varibles. Given a set of observations consisting pairs of $X,Y$, is it possible to attach it as probabiltiy density ...
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16 views

Moment generating function of quadruple-form of Gaussian RVs

Let $X \sim N(0, I_d)$ be a $d$-dimensional Gaussian random variable. Let $\beta_1$ and $\beta_2$ be two $d$-dimensional vectors. I would like to compute expectation \begin{align} \mathbb{E}\Bigl \{ ...
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19 views

Using the normal approximation, what is the $z$-value of a sample difference of $\hat p_1−\hat p_2=−0.18$? [closed]

A few concepts from my textbook that I do not understand: If the Random, Normal, and Independent conditions are met, then is it true that $\hat p_1+2\hat p_2$ is approximately normally ...
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18 views

how to determine the expected value of the first element of an ordered statistic

let $X$ be a random variable with density $$f_x(x) = e^{\alpha-x} \: x>a$$ (a) what is the maximum likelihood estimator for alpha (b) make the estimator in (a) unbiased, and evaluate its ...
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40 views

Estimate of Proportion

An airline is interested in determining the proportion of its customers who are flying for reasons of business. If they want to be 90 percent certain that their estimate will be correct to within two ...
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58 views

Obtain a $(1-\alpha) 100$% confidence interval for $\theta$ using the moment estimator

Suppose $x_1,..x_n$ is a random sample from a distribution with probability density $f(x|\theta )=\theta x^{\theta -1} 0<x<1 \ and \ \theta >0.$ Find the moment estimator of $\theta$ and ...
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16 views

One-sample t tests for sample mean with outliers

In carrying out a one-sample t test for a sample mean, how should outliers be dealt with? For example, for the data ${110, 110, 110, 118, 122, 150}$ in a sample size of 6, evidently $150$ is an ...
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9 views

How to find the effects of the parameters with the Latin Hypercube Simulation?

We are working with an epidemic model where we have 21 parameters which affect the $R_{0}$-value. We want to study the effect of those parameters on the $R_{0}$-value with the Latin Hypercube ...
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30 views

Why this equation holds, using Bayes formula?

My mysterious equation is: $$p(x|\chi)=\int_{\theta\in\Theta}p(x|\theta)p(\theta|\chi)d\theta$$ where $\chi$ is some samples drawn from sample space parameterized by $\theta\in\Theta$. Follows the ...
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1answer
36 views

Correctly calculating the bias of an estimator

I'm currently learning about method of moments and maximum-likelihood estimators and have confused myself with this issue: First, let me estimate the parameter $\lambda$ from the exponential ...
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27 views

Confidence intervals and significance tests

A sample is used both to construct a 95% confidence interval for a population proportion p and to run a significance test with null hypothesis $H_0:P=0.07$ and significance level $\alpha=0.05$. Is it ...
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10 views

Additive model ANOVA with interaction. How to obtain gammas from the graph?

I try to complete this exercise of statistic that said that I need to compute the values of alpha, beta and gammas from the graph. Graph: I allready compute the values algebricaly, and this are: ...
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11 views

How to prove the ancillary statistic problem?

Let $X_1, ...,X_n$ be a random sample from the following pdf: $f(x) = e^{-(x-\theta)} \exp(e^{-(x-\theta)}), -\infty<\theta<\infty$ a. Define $W=\log(\sum_{i=1}^{n}e^{-X_i})$.Shoe that ...
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13 views

Distribution of Subvector Sums

Suppose $X_1,\dots,X_N \sim_{iid} \mathcal{N}(0,1)$ are iid normal, and let $K=N/2$. Let $S$ denote the collection of all subsets of $\left\{1,\dots,N\right\}$ with $K$ elements. For any $s\in S$ ...
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18 views

estimating $\mu$ of a poisson random variable using method of moments or maximum likelihood

A used car salesman is willing to assume the number of sales he makes, per day, is a Poisson random variable with parameter $\mu$. Over the past 30 days, he made $0$ sales on $20$ days and one or ...
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1answer
23 views

Simple random sample of a Bernoulli and probability function of a statistic.

Let $x_1, \ldots, x_n$ an simple random sample of a Bernoulli distribution of parameter p, $0<p<1$. Let $\overline{\textbf{x}}$ be the mean sample and $S^2=\frac{1}{n-1}\sum_{i_1}^n ...
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20 views

For $X = (X_1,X_2,…,X_n)$ i.i.d. $N(0,\theta),$ show $ T = \bar X $ is not complete for $\theta$.

For $X = (X_1,X_2,...,X_n)$ i.i.d. $N(0,\theta),$ show $ T = \bar X $ is not complete for $\theta$. Definition: Let f (t, θ), θ ∈ Θ be a family of distributions for a statistic T (X). The family is ...
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67 views

Non-Linear Model Transformation

I want to transform this Non-Linear Model $y= 8-ae^{bx}$ to Linear.And my issue is in this step $lny=ln(8-ae^{bx})$ how can simplify it to reach in a linear model which is like this $y*=b0+b1x$ ...
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30 views

expected value of random variable times sample mean

Assume a population of $M$ numbers, where $M_i$ have values $y_i$, $i = 1,2,...,k$; thus $\sum_{i=1}^k{M_i} = M$. Thus the population mean is $\mu = \sum_{i=1}^k{\frac{M_iy_i}{M}}$ and the ...
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21 views

Can I bound $P[R > x + \epsilon]$ independently of R?

I have this probability distribution: $P[\Theta < \varphi] = \frac{\varphi}{\pi}$ for $\phi \in [0,\pi]$. Now I have $n$ samples of $D = R\Theta$ i.i.d. ($R>0$) and I want to estimate $R$ as ...
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28 views

Definition of Power and relationship with Type II error

I've seen two definitions of Power: $P(\text{Rej. } H_0|\theta \in \Theta_1)$, from Wiki. $P_{\theta}(\text{Rej. } H_0)$, from Casella and Berger 'Statistical Inference'. Which one is true? If we ...
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17 views

Statistics for Sports League Qualification

How can one quantify and predict the needed points for qualify in a league given an up-to-date results registry? For instance, regarding Basketball Euroleague, there's 8 teams in a league with direct ...
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1answer
18 views

confidence about range of standard deviation

Suppose $X_1, X_2$ is a random sample of size 2 from a normal population known to have mean $0$ and variance $\sigma^2$; further assume $x_1 = -0.75$, $x_2=0.16$. How sure (or confident) would you ...
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21 views

How do I show that the two methods of permutation test are both the same?

My main objective is to show the methods described below are really the same. However, I am having difficult both formulating the idea clearly and proving my assertion. Below is my attempt. Suppose ...
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27 views

Why can we simply pool the realized observations in a permutation test?

Let a vector of i.i.d random varibles $(X_1,X_2,X_3,\cdots, X_m)$ and another vector of i.i.d rvs $(Y_1,Y_2,Y_3,\cdots, Y_n)$ be given. Suppose $X_i$ stands for the recovery time using a new treatment ...
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15 views

Singular matrix in GMM based on variational Bayes computation

I have implemented some codes according to the variational inference here. In the code $W_k$ is computed according to $$ W_k^{-1}=W_0^{-1}+N_kS_k+\frac{\beta_0N_k}{\beta_0+N_k}(\bar x_k-\mu_0)(\bar ...
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1answer
20 views

Residual plot in the logarithmic model.

We've got some data containing two variables, where $x$ is the predictor and $y$ is the response variable. We make a model of the form of: $$y=\alpha+\beta \cdot x + \epsilon$$ Then we see that in the ...
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26 views

Bayesian inference for sum of random variables

Assume that we have a random variable $Z = X + Y$ for $X$ and $Y$ independent. Then if w use two independent data-sets $D_1$ and $D_2$ to try and approximate the distribution of $Z$, i.e. ...
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1answer
18 views

Why aren't the degrees of freedom included in the confidence interval formula?

I have one question. In the formula for calculating the confidence interval, I don't see the degrees of freedom in the formula. In my book, the formula is : $$\text{Confidence interval:} \qquad \mu ...
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38 views

Given the table, calculate the table of joint probability of $A$ and $B$.

Given the below table, the sample consisted $20\%$ of $B_1$, $20\%$ of $B_2$ and $60\%$ of $B_3$. Calculate the table of joint probability of $A$ and $B$. $$\begin{array}{|l|l|l|l|} ...
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Do unbiased estimators have to be exactly equal to the true value of the parameter?

Is it true that for an unbiased estimator, the mean of the sampling distribution is very close to, but not always equal to, the true value of the parameter being estimated? My textbook says that "An ...
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1answer
49 views

Rejection region

I need help with the following problem: Let H0: p = 0.6 HA: p = 0.7 based on observing a binomial random variable with 10 trials. What is the rejection region for the most powerfil level sigma ...
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1answer
25 views

Distribution of parameter under the null hypothesis for mixture distributions

I am conducting a classical hypothesis test concerning the value of some parameter, i.e. $H_{0}:\theta=\theta_{0}$. I'll denote the distribution of $\theta$ under the null as $f(\theta)$. Suppose ...
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36 views

Test if the theory fits the data

We are given that the members of a community are classified by blood type according to the following schema: \begin{array}{|c|c|c|c|c|} \hline O& A & B & AB & Total \\ \hline 121 ...
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7 views

Interpretation of statistical significance

I have some experimental data, not related to the field of statistics, but I need to verify whether if follows a certain distribution or not. After a quick research I found the Chi squared test to be ...
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2answers
34 views

If $Y_1, \ldots, Y_n \sim Poisson(\lambda)$ are iid, how to show that $E(Y_1 | T = \sum_{i=1}^{n}Y_i) = \frac{T}{n}$?

Suppose I have that $Y_1, \ldots, Y_n \sim Poisson(\lambda)$ are iid. I saw a line in a book that said that if $T = \sum_{i=1}^{n}Y_i$, then: $$ E(Y_1 | T) = \frac{T}{n} $$ I am lost as to how they ...
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23 views

Sufficient statistic for Bernoulli r.v.

Let $X_1,X_2,X_3$ be a iid sample of the Bernoulli ($p$) distribution. Consider the statistic $T = X_1X_2 + X_3$. Show that $T$ is not sufficient for p. I did ...
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16 views

Relation between Bayesian analysis and Bayesian hierarchical analysis?

I have been studying a Bayesian hierarchical model. In that model all I am dealing is with the estimation of parameters. In Bayesian analysis, loosely speaking, we update our prior knowledge (in light ...
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41 views

Confidence interval for exponential - is it the shortest possible?

The confidence interval for an exponential distribution is said to be: $$\frac{2n\bar{x}}{\chi^2_{1-\alpha /2,2n}}<\frac{1}{\lambda}<\frac{2n\bar{x}}{\chi^2_{\alpha /2,2n}}$$ In general we aim ...
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1answer
34 views

Confidence interval; exponential distribution (normal or student approximation?)

Let's say we have got a sample of size $n$ from an exponential distribution with an unknown mean $\lambda$. We want to construct a confidence interval and so we can compare this: ...
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1answer
27 views

Normal distribution: likelihood ratio and rejection region for estimating sigma

I'm having a little trouble with the following problem: "Let $X \sim N(0,\sigma^2)$ and consider testing $H_0: \sigma = \sigma_0$ versus $H_A: \sigma = \sigma_1$, where $\sigma_1 > \sigma_0$. The ...
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18 views

consistency and law or large numbers

I was curious if there is a relationship between consistency and Weak Law of Large Numbers (W.L.L.N.). In the sense that to apply the W.L.L.N., an estimator for the mean $\mu$ must be consistent. I ...
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24 views

how to compute p-value when the confidence interval is given?

Please help on this problem...This is the statement: "The immunological assay verified the presence of RuBisCO in non-treated (control) and Pb-induced leaves with average relative band intensities of ...
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39 views

what is the expected value of 2 raised to the power of a random variable $X$

I'm trying to figure out how to calculate the variance of some data I have. The basic throughput of the analysis is as follows: 1) Get data in triplicates for each group plus a control group & ...
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2answers
35 views

MLE of double exponential

I am given the double exponential distribution under the form $$f(x_i\mid\theta) = \frac{1}{2}e^{-\frac{1}{2}|x_i - \theta|}$$ and I need to find the MLE of $\theta$. I have two approaches until ...