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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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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Pivot for exponential

Supose that $X_{1}, X_{2}, \ldots , X_{n}$ are i.i.d. exponential with $\lambda$ parameter. Show that $2\lambda X_{i}$ have a ji square distribution with 2 freedom degree. And $2\lambda \sum X_{i}$ ...
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18 views

why does exact binomial confidence intervals have wider than nominal coverage?

The exact $(1-\alpha)$ level confidence interval lower limit is given by $$ \sum_{k=y}^{n} {n \choose k} {p_L}^k(1-p_L)^{n-k}=\alpha/2 $$ and the upper limit analogously. Why does the resulting C.I. ...
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1answer
52 views

Transformation of a Random Variable

We have a random variable $x$ with p.d.f. $\sqrt{\dfrac{\theta}{\pi x}}\exp(-x\theta)$, $x>0$ and $\theta$ a positive parameter. We are required to show that $2\theta x$ has a $\chi^2$ ...
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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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10 views

Bayesian Uni-variable ou multi-variable and formulation

I have a parameter that has a prior distribution with mean equals to 30, a variance of 25 and a number of samples $n=30$. I was able to execute 30 more samples, and I got a mean of 25 and a variance ...
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2answers
12 views

Population Variation with two variables

I have a dataset with two variables. I want to treat my dataset as a population not a sample. I am wondering if I can just use the formula for population variance as below: ...
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55 views

Question about piecewise exponential distribution

This is an excerpt from the following paper. I am particularly interested in knowing how the authors got the displayed equations. We let [Z] denote the distribution of a generic random variable ...
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53 views

Difficulty to compute an integral

Have somebody ideas to evaluate the following integral ? $$J_n=\int_{-\infty}^{+\infty} \left(\frac{\pi^2}{4}-\arctan(x)^2\right)^n\,dx$$ I'm trying this because I have shown that the empiric ...
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1answer
22 views

Practical examples of same mean and different variance

I recently came across this paper : http://projecteuclid.org/download/pdf_1/euclid.aos/1176343959 It proposes a theory about estimating the common mean of several normal populations(which have same ...
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1answer
23 views

Distribution under null-hypothesis and type 1 error

Given random variables $X_1,...,X_n \overset{i.i.d.}{\sim} N(\mu, \sigma^2)$ where the variance $\sigma^2$ is known let the null hypothesis be $H_0: \mu = \mu_0$ For the statistic $T=\sum_{i=0}^nX_i$ ...
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25 views

Statıstıc problem

Will I use binomial distribution for this question? Can you help me please thnk you
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20 views

Statistic problem

Can you help me to solve this problem pls,I have exam and I am studyıng. What wıll I use, bınomial or Other thing ? Thank you
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1answer
27 views

Normal approximations and Binomial distributions

I am having some difficulty with the following question from my textbook. I have really been trying to understand the use of normal and binomial approximations, but I'm getting really confused. Any ...
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1answer
15 views

Finding a confidence interval given sample size, mean, and standard error

I am not the best with Statistics, and I was wondering if it is possible (and how, if it is) to find a 93% Confidence Interval given Sample Size (27), Mean (6.73), and Standard Error (1.732). Thank ...
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1answer
30 views

Find the bias and MSE of an estimator for the upper limit of a uniform distribution.

Let $X_1,\ldots, X_n \sim U(0,θ)$ and let $$\hat{θ} = 2\cdot\dfrac{1}{n}\sum_{i=1}^{n}X_i$$ Find the bias, se, and mse of this estimator. I saw a similar question, but I can't figure out how to get ...
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1answer
63 views

How to prove a statistic is not complete

Suppose X is a Poisson($\lambda$), where $\lambda\in\{0,1,2,...\}$, how to prove X is not complete. It seems like that we need to find a function $g$ which is not identical $0$, such that the ...
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18 views

Approach on Solving a Interence problem using CLT

What is the probability of finding people in U.S having weight greater than 160lbs. This is what I am thinking to do. I sample 40 people randomly, get their mean and repeat this sampling 100 times. ...
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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 ...
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1answer
25 views

Central limit theorem applicability

Couple of doubts: 1) The CLT requires you to have population distribution and population parameters before it can you used. Correct ? It cannot be then used to solve problems where getting an entire ...
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37 views

Cumulative distribution function of exponentials

I have the cumulative distribution function $F(x)=(1-e^{-x})\mathbb{1}_{x≥0}$ and want to write the CDF to $F(\frac{x-\mu}{\sigma})$. I have derived ...
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45 views

Statistical Inference, Differential Geometry and Entropy

Context: Statistical Inference and Differential Geometry Let's consider a generic $ p(x;\theta) $ distribution with $ \theta $ Parameters Vector, it is obvious that $$ \int p(x; \theta) dx = 1 $$ ...
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16 views

Contingency table with a 0 value in a concrete problem.

I don't know how to deal with this problem: In order to evaluate the relationship with a risk factor and a disease we have the following case-control study: \begin{array}{|c|c|c|} \hline Risk Factor ...
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1answer
35 views

Statistical Inference and Manifolds

I have just begun approaching the connection between statistical inference and differencial geometry. If I got it correctly, one of the most fundamental concept regards the connection between a $ ...
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1answer
36 views

asymptotic normality and unbiasedness of mle

Suppose $\hat{\theta}_n$ is the MLE for some parameter $\theta$. Suppose also that the MLE is such that the Cramer regularity conditions are fulfilled, and $\hat{\theta}_n$ is asymptotically normal ...
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1answer
51 views

exactly k consecutive heads, n tosses

What is the expected number of strings of exactly k consecutive heads if a fair coin is tossed n times? My current answer is $$ {n-1\choose k} (\frac{1}{2})^{(k-1)} $$ Is this correct? A possible ...
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26 views

Expected number of sides of a dice

I have two dice, one with m sides (labeled 1,2,...m) and one with n sides (labeled 1,2,...n). I roll both three times. The m-sided one comes up 1, 2, 9 and the n-sided one comes up 7, 7, 8. Which is ...
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19 views

Interpreting confidence interval of regression coefficient.

In a Simple Linear Regression analysis, independent variable is weekly income and dependent variable is weekly consumption expenditure. Here $95$% confidence interval of regression coefficient, ...
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46 views

Population estimate from sample

This seems very basic but I can't find a clear statement of it. Suppose I have a population of N balls which are red, white, and blue in some proportion. If I take a sample of S balls (S << N) ...
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58 views

Linear Algebra question relating to eigenvectors

Let A be an m x m positive definite symmetric matrix with eigenvalue-eigenvector pairs $(\lambda_1,e_1),....,(\lambda_m,e_m).$ The eigenvectors are orthonormal. Let $C = e_1e_1'+....+e_me_m'$. ...
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15 views

Question regarding Balanced Incomplete Block Design

Question: Consider a BIB design with a treatments, b blocks and c < a number of plots in each block where a,b,c ≥ 2. Let $n_{ij} = 1$ if an observation is made on the ith treatment in the jth ...
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MLE of $n(\theta,a\theta)$ family

Question: A special case of a normal family is one in which the mean and variance are related, the $n(\theta,a\theta)$ family. If we are interested in testing this relationship, regardless of the ...
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Whats the reason of log-odds?

I'm following a course of bayesian inference. Out of the middle they are talking about log odds. For example, say that we have a uniform prior distribution, $\theta$ ~Beta(1,1). $$g(\theta) = ...
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66 views

Calculate the PMF, mean and variance of X for x=-1,1

An Urn contains 7 red and 11 white balls. Draw one ball at random from the urn. Let X=1 if a red ball is drawn, and let X=-1 if a white ball is drawn. Give the pmf, mean, and Variance of X. I know ...
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1answer
41 views

Is it possible to find an asymptotic distribution for the likelihood ratio test without the maximum likelihood estimators being consistent?

The usual proofs of the asymptotic distribution of the likelihood ratio test (LRT) being a chi-squared assume that the maximum likelihood (ML) estimators are consistent. Is it possible to find an ...
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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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1answer
20 views

Prros and cons of including controls in a regression?

Assume we have conducted a random experiment for the benefits of a drug. Let $Y_i$ be the outcome of interest , $X_i$ be some control variables (e.g. age, sex etc.) and $$D_i= ...
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51 views

Prove $\operatorname{Var}(\hat{e}_{ij}) = \sigma^2 \left(\frac{n_i-1}{n_i}\right)$

$\newcommand{\Var}{\operatorname{Var}}$ Let $y_{ij}$ denote the observed response of the $j$th experimental unit in the $i$th treatment group, and the $e_{ij}$ are i.i.d. $N(0,\sigma^2)$ experimental ...
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Subset of samples has any effect on sufficiency of the statistic?

If we have the following iid samples $$ X_1, ..., X_n \sim N(\mu, \sigma^2) $$ where only $\mu$ is unknown. We know one sufficient statistic is the following: $$ T = \frac{1}{n} \sum_{i=1}^n X_i $$ ...
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Error type I for $X_i \sim Exp(\theta)$

Let $ X_i$ be i.i.d $Exp(\theta)$ for i=1,...,4.We want to test $H_0: \theta =6$ versus $ H_1: \theta = 2$. Consider the following test: $$\text{Test: Rejects H_o} \iff \frac{X_1 + X_2}{2}>4.5$$ ...
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45 views

Find $P${$min_{i \neq j}|R_i-R_j| \geq d$}, where $R_1,…,R_n$ are uniform on line with length L

If n points $R_1,...,R_n$ are picked independently and with uniform density on a straight line of length L, find the probability that no two points will be less than distance d apart; that is, find ...
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52 views

Probit regression model: Construction of weighted least squares algorithm

I'm posting a difficult general linear model question which I would like to solve. Question: Consider a probit regression model for $y \in ${$0,1$}:$E(y|x)=\Phi(x'b)$, where $\Phi$ is the standard ...
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1answer
53 views

Sample size required to estimate population proportion with given precision

It plans to conduct a study on the percentage of homeowners who have at least two TVs. What should be the sample size if we want to ensure that $95\%$ of estimation error is less than $0.01$? ...
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Bivariate distribution with normal conditions

Define the joint pdf of $(X,Y)$ as: $$f(x,y)\propto \exp(-1/2[Ax^2y^2+x^2+y^2-2Bxy-2Cx-Dy]),$$ where $A,B,C,D$ are constants. Show that the distribution of $X\mid Y=y$ is normal with mean ...
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22 views

Ratio of hazards in Proportional Odds model

In the proportional odds model we have the the odds of survival in 1 group are proportional to the odds of survival in another group $$\dfrac{ S_1(t)}{1-S_1(t)} = \psi \dfrac{S_0(t)}{1-S_0(t)}$$ ...
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19 views

Joint Limiting Distribution of Min and Max

Let $X_1,\ldots,X_n$ be iid from the uniform distribution $U(a,b)$. Let $X_{(1)}< cdots< X_{(n)}$ be the order statistics. Find the joint limiting distribution of $(n(X_{(1)}-a),n(b-X_{(n)}))$ ...
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140 views

How do I find the PMF of X when X is the number of flips of a fair coin that are required to observe the same face on consecutive flips?

How do I find the PMF of $X$ when $X$ equals number of flips of a fair coin that are required to observe the same face on consecutive flips? The hint was to draw some sort of a tree diagram, but I'm ...
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6 views

Show that $Y = 2\sqrt{X_1 X_2}$ has a $\Gamma(2p, 1)$ dist.

$X_1$ and $X_2$ are independent with $\Gamma(p, 1)$ and $\Gamma(p + 1, 1/2)$. Show that $Y = 2\sqrt{X_1 X_2}$ has a $\Gamma(2p, 1)$ dist.
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Sum squared errors normal

Let $X_1,..,X_n$ be independent normal random variables with common variance $\sigma^2$ and means $a+bc_i$ (where $a,b,\sigma^2 $ are constants $>0$). If $s_1,s_2$ are real numbers minimizing ...
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Predicted value of polynomial regression models

Suppose that we have a polynomial linear regression as following $$ Y_i = \beta_0 + \beta_1 X_{i} + \beta_2X_{i}^2 + \epsilon_i, \quad i=1,\ldots, n $$ with $\epsilon_i \sim N(0,\sigma^2)$ and ...