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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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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24 views

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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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
24 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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35 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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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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14 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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40 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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30 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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25 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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1answer
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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35 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
31 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 ...
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49 views

Nonparametric Estimation of the Hazard Ratio

I am not sure if this is appropriate here, but I'm hoping someone would be able to help. The following is an excerpt from the following paper "Nonparametric Estimation of the Hazard Ratio" by ...
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34 views

What is the relation between Bayes' theorem and Gibbs distribution.

This question refers to the link: https://en.wikipedia.org/wiki/Principle_of_maximum_entropy specifically to the last sentence of the page: "For the case of given average values as testable ...
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44 views

Bayesian inference exercise

I am learning online Bayesian Statistics and I have a test in a couple of days. I have no idea how to solve this exercise, any help will be appreciated. There might be something similar in the quiz... ...
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27 views

sufficiency of linear combination of bernoulli random variables. [closed]

Let $ X_1$, $X_2$, $X_3$ be a set of three independent Bernoulli random variables with unknown parameter $p$ = $P(X_i=1)$. Where it is given that $p$ = $X_1 + X_2 + X_3 $ is sufficient for $p$. Show ...
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Hypothesis testing: 2 groups with 2 subgroup each. Which test to use?

I recorded the pumping rate (measurement variable) of two types of worm: normal and mutant, with n=6 the number of worms per type. Then, I exposed these types of worms to a drug. BUT I used a ...
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1answer
23 views

Can I apply a one-way ANOVA and Tukey test to this data?

The experiment: Effect of pH on yeast respiration. I tested 5 different pH levels and measured the amount of CO2 produced. The data shows a trend similar to upside down parabola. Would it make sense ...
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1answer
34 views

Statistics: Random Variable within a Random Variable

Say I have two uniformly distributed random variables as follows: $X$ ~ $U(a,b)$ $Y$ ~ $U(0, X)$. How would I go about finding the expectation of $Y$. Obviously the expectation of $X$ is simple, but ...
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Deriving posterior from prior

If a signal is assumed to evolve in time like the OU process $$ dX_t = -a X_t dt + \sigma dV_t $$ with observations $$ dY_t = h X_t + dW_t $$ I am told that, the posterior distribution, $\pi_t = ...
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Sampling from Gaussian Process Posterior

Anyone know of a Python package that both fits a Gaussian Process to data, and also lets you sample paths from the posterior? I'm interested in sampling the colorful lines on right (b) of the ...
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13 views

Example for unknown parametrs chosen in Bayesian and frequentist inference?

The difference between Bayesian and frequentist inference is that in Bayesian analysis, parameters are random but in frequentist analysis they are fixed but unknown. Can someone explain to me this ...
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1answer
13 views

Moment expansion of likelihood function

Let's say I have a binary hypothesis test described by a probability distribution $p(x;\theta)$ where $\theta$ takes two different values under each of the two hypothesis ($\theta_0$ under ...
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27 views

Recognize elements of an hypothesis testing problem

Before the launch of a commercial product, a company makes a market survey to know the price that buyers are willing to pay. It is assumed that this price is normally distributed with a ...
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23 views

How do you interpret a scatterplot distribution with a gap?

I want to see if there is a relationship between sample area size and number of animals caught. I have never encountered a scatterplot with a gap like this before.
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27 views

Likelihood Ratio Test for Exponential Distribution with a Limited Parameter Space

Suppose that we are given an exponential distribution model with a pdf $f(x,\theta) = \theta^{-1}\exp(-x/\theta)$ with an iid sample $X_1, \ldots, X_n$, and we would like to test hypothesis $H_0 : ...
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79 views

Sampling with replacement or without replacement

I'm writing a program in R that simulates bank losses on car loans. Here is the questions I'm trying to solve: You run a bank that has a history of identifying potential homeowners that can be ...
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28 views

Weird behaviour of CLT's application to binomial.

I am carrying out the simulations of the following experiment for all $n$ in the set $\{1,2,3,...,100\}$. (0) Set $k=0$. (1) Generate $n$ $Bernoulli(0.9)$ trials. (2) Construct estimate ...
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The definition of Odds Ratio isn't helping me understand this calculation.

I am trying to understand an Odds Ratio calculation:cross table calculation for OR I did the calculation and calculated $.458$ as the OR. The statistician calculated $2.1$. I am thinking that he ...
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1answer
40 views

Probability density function and the minimal sufficient statistics for two samples from normal distribution

Suppose $X_1,\ldots, X_m$ is a random sample of size $m$ from the normal distribution $N(\mu_1,\sigma^2)$ with mean $\mu_1$ and standard deviation $\sigma$, and that $Y_1,\ldots, Y_n$ is a random ...
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MLE estimator of $\theta$ for $f(x;\theta)=e^{i\theta -Xi}$

$X_1...X_i$ are independent randomly distributed with pdf $$f(x_i|\theta)=e^{i\theta-x_i}$$$ $$x_i≥i\theta$ -Previous problems I had done where the range of x was dependent on $\theta$ just involved ...
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46 views

Proof: Pivotal Quantity

Can anyone give me a clue of how to address this theorem? Suppose that $T$ es a real-valued statistic. Suppose that $Q(t,\theta)$ es a monotone function of $t$ for each value of $\theta\in ...
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41 views

Estimation of probability distribution for alternating renewal process

I am studying so-called alternating renewal process, and I have a question concerning estimation of on-time/off-time distributions. Settings Let $\{(Z_0,Y_0),(Z_1,Y_1),(Z_2,Y_2),...\}$ be a sequence ...
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1answer
59 views

Hypotheis Testing (paired T test): 1) Is my work correct? 2) How to graph in SPSS,?

I have the following Hypotheis testing problem. The statement of the exercise is: Experiment: Eleven different varieties of barley were considered. Of each variety, half was kiln-dried and the ...
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1answer
13 views

Hypothehis test (Z- two simple proportion). I'm try to understand from where came this numbers.

I'm trying to find out from where came some proportions in a hypothesis test problem from my class notes. Here is the problem and tables: Statement "X" Pm= the proportion of men who agree with ...
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18 views

how we get the penalty term in logistic regression likelihood

When the likelihood of logistic regression is $$\ \ell(\beta;y)=\sum_{i=1}^{n} \pi_i^{yi} (1-\pi_i)^{(1-y_i)}$$ And if we have a model with random parameter, then the likelihood will be $$\ ...
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1answer
28 views

Show that sample variance is unbiased and a consistent estimator

I am having some trouble to prove that the sample variance is a consistent estimator. I have already proved that sample variance is unbiased. I understand that for point estimates T=Tn to be ...
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27 views

How could an estimator be biased but consistent according to mathematical definition?

According to the definition, an estimator can be biased, if $E_{\theta}[\hat{\theta}]\ne\theta$, with $\theta$ as parameter for a distribution we want to get from samples. While the estimator can be ...
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1answer
29 views

Jensen's inequality in derivation of EM algorithm

I am going through the derivation of EM algorithm and got stuck on understanding the following steps: Notes showing EM algortithm derivation For the equality to hold, f(x) has to be an affine ...
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90 views

How to compute the $p$ value? and the correct explanation of the overall experiment.(Is my answer correct?)

Hello community first of all thanks for helping me with my math problems. Here I'm again with hypothesis test exercise. I want to know if I made some mistake in my answer and if someone can help me ...
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1answer
54 views

finding UMVUE $P ( X_1> t)$ [closed]

Suppose $ X_1,\ldots,X_n$ are i.i.d. random variables with density: $$f(x_i;\theta)=\theta x_i^{-2}$$ $$x_i>\theta$$$$\theta>0$$ The smallest order statistic $X_{(1)}$ is sufficient and ...
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1answer
23 views

How to derive mean and variance for a Bayes estimator?

Let $X_1,...,X_n \sim$ iid $\mathcal{N}\left(\theta , \sigma ^2\right)$, where the variance is known. Also, suppose the prior distribution $\theta \sim ...
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1answer
22 views

Proving Weak Law of large numbers by Markov's inequality

Hi I am trying to solve the problem 5.13 of the book Statistical inference by George Casella and Roger L. Berger. The problem is Formulate and prove a version of the WLLN with a weaker assumption ...
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14 views

Estimate ratio of two expectations by sample means

I have a question about the estimation of a ratio of two expectations. Suppose $X_{i}$ and $Y_{i}$ are two random variables with $i=1,\cdots,N$. We seek to estimate $\mathbb{E}X_{i}/\mathbb{E}Y_{i}$ ...
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1answer
42 views

Proving that a statistic is not sufficient (uniform case).

Let $X=(X_1,...,X_n)$ be i.i.d. $U(0,\theta)$. How to show that $$\frac{2}{n}\sum_{i=1}^{n}X_i$$ is not a sufficient statistic? I have already proven that $\max_{i=1,...,n}X_i$ is a sufficient ...