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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Statistical sample with age ranges. How to extrapolate it using the real age distribution over population.

I have a data set consisting in the classification of the numbers of suicides by age range. I want to figure out if there is or not association between the number of suicides and the age range. But, ...
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22 views

Why does $\int f(x)(y-r(x))\;dP(y,x) = 0$?

My question is, why does: $$\int f(x)(y-r(x))\;dP(y,x) = 0,$$ where $r(x) = \int y \;dP(y|x)$ and $P$ is a probability distribution function. It was also given (in my book) that: $$\int ...
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24 views

Which model to be used for predictive analysis [on hold]

I have a problem where i have been given set of data against month example Month | Data1 | Data2 1---------5--------5 2---------6--------7 Consider the data 1 be the temperature and data 2 be the ...
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66 views

$\mathsf kth$ moment of the standard deviation about the origin from a $\mathsf N(\mu,\sigma^2)$ population

Let T be the standard deviation of a random sample of size n from a $\mathsf N(\mu,\sigma^2)$ normal population. Find the $\mathsf kth$ moment of T about the origin, and state the condition for the ...
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12 views

One-to-one correspondence between mean value and parameters

I am currently taking a course in statistics, and in this course we are considering linear models $\mu = X\beta$ where $\mu \in L$ and $L = col(X)$ is a linear subspace of $\mathbb{R}^n$, $X$ is the ...
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Simulate from a distribution using Metropolis-Hastings and Rejection Sampling?

We have covered the basics behind rejection sampling as well as Metropolis-Hastings from class, but I am not sure how to use the two in conjunction to solve the following problem: Given $\pi(x) = ...
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8 views

Find the UMP test for Hypothesis testing

Let $X_1,...,X_n$ be a random sample from the $N(μ,σ^2)$. Assume $μ=0$ (a) Find the UMP test for $H_0 : σ^2=σ_0^2$ versus $H1 : σ^2 < σ_0^2$ at signicant level $\alpha$. (b) For the UMP test in ...
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Asymptotic confidence interval

Let x1, x2, ..., xn be a random sample with a density function given by $ f(x) = \frac{3}{\theta^3} x^2 I_{(0,\theta]}(x)$ where $I_{(0,\theta)}(x)$ is the indicator function and $\theta > 0$ ...
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12 views

How to arrive at this conditional expectation $E[Y|A]= E[Y] + \frac{Cov(A,Y)}{Var(A)}(A-E[A])$

My professor arrived at the formula (1) but I cannot figure out how he got there. Is (1) from an identity after some manipulation? The setup is: X,Y are correlated, jointly gaussian random ...
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15 views

Boxplot comparisons

How do I compare the following boxplots? Comparing locality; I would say that, the median of the second boxplot (to the right) is higher than the median of the first boxplot (to the left). ...
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optimising likelihood function in R

I want to try and optimise the function below in R to find best estimates of $\tau^2$ and $\mu$. $L(\mu, \tau^2) = -\frac{1}{2}\sum\limits_{i=1}^k \mbox{log}(2\pi(\sigma_i^2 + \tau^2)) - ...
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22 views

How can I uniformly draw points from an ellipsoid?

Specifically, given a positive definite matrix $A \in \mathbb{R}^{n \times n}$, how can I efficiently generate points $x \in \mathbb{R}^n$ that satisfy $x^TAx \leq 1$? I know how to do this when the ...
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12 views

Find a complete sufficient statistic

Here is my problem : Suppose theta is a nonrandom parameter satisfying theta > 1. Suppose further that, given theta, Y1 , Y2, ... , Yn are i.i.d. observations with each density f_\theta(y) = (\theta - ...
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36 views

Indepenent variables and functions

Random variables $x_1, x_2,...,x_n$ are independent. Then, how to prove whether these functions $$y_1=f_1(x) \\ y_2=f_2(x) \\ ... \\ y_n=f_n(x)$$ are independent or not . where, $x=(x_1,...x_n)$ ...
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4 views

Effect of coding technique used for qualitative variables, on the variance

In regression modeling, can using a certain coding technique for qualitative variables, be better than others. As in, say can using effect coding instead of dummy coding it reduce the total Error sum ...
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10 views

Sufficient statistics for $\mu_1 -\mu_2$

If $ X_1, ..., X_n$ is a random sample from $ X \sim N(\mu_1, \sigma^2)$ and $Y_1,..., Y_n$ is a random sample from $Y \sim N(\mu_2, \sigma^2),$ if the samples are independent and $ \sigma^2$ is ...
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18 views

Exponential distribution word problem involve watch

The amount of time that a watch will run without having to be reset is a random variable having an exponential distribution with $\theta=120$ days. Find the probability that x is the amount of time ...
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20 views

How to account for both between-subjects and within-subjects covariables?

I have a data set I'm trying to analyze and I can't figure out how to include the two different kinds of covariables in my analysis. Without the covariables, the analysis isn't too complicated. It's ...
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23 views

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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24 views

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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1answer
16 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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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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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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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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53 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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51 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
19 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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20 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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24 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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25 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
12 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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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
60 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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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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19 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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36 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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44 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
34 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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34 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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47 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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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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Mathematics Model for measuring the evenness of a distribution

At time $t$, the distribution for a dynamical model is: $a_1(t), a_2 (t), a_3 (t),…, a_n(t)$ as the system evolves it may be expected that if the number of samples in a species is less than the ...
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16 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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43 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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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 ...