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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Likelihood Function for the Uniform Density.

Let the random variable $X$ have a uniform density given by $$ f(x;\theta)=I_{[\theta-\frac{1}{2},\theta+\frac{1}{2}]} $$ where $-\infty\leq\theta\leq\infty $ the likelihood function for a sample of ...
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43 views

Finding P value

I have these observations $(2,3.2,3.8,2.5,3.3,2.8,3.0,3.4)$ from $X \sim N(\mu,\sigma^2)$ and i want to calculate the $P$-value testing $H_0: \mu =3.2$ against $H_1 \neq 3.2$ with $\sigma = 0.6$ ...
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showing that $E[(\hat\theta -\theta)^2] \lt Var(\bar X)=\dfrac{1}{n}$. [closed]

Suppose $X_1, X_2, \dots, X_n$ are i.i.d $N(\theta, 1),\theta_0 \lt\theta$ , Find the MLE of $\theta$ and show that it is better than the sample mean $\bar X$ in the sense of having smaller mean ...
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Showing independence of random variables

When proving $\bar x$ and $S^2$ are independent in my noted it says that "functions of independent quantities are independent ". Can someone tell me how functions of independent quantities are ...
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Distribution of the sample variance

This is my first post to this great website :) It seems like an excellent place to learn. I have a question however that is bothering me as I cannot figure it out through my textbook. The sample ...
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179 views

How to find the following integration

Let $X_1, \cdots, X_n$ be $iid$ normal random variables with unknown mean $\mu$ and known variance $\sigma^2$. How to find $E[\Phi(\bar X)]$, where $\bar X:=\frac{\sum_{i=1}^nX_i}{n}$, please? I guess ...
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271 views

Sampling error with weighted mean

I am studying statistics and I am wondering when it comes to standard error or a sampling if the calculation changes when there are weights added. I have a weighted mean: $$\mu_{w} = ...
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Maximum Likelihood Estimator for Multinomial.

Suppose that 50 measuring scales made by a machine are selected at random from the production of the machine and their lengths and widths are measured. It was found that 45 had both measurements ...
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141 views

Show $\psi$ and $\Delta$ are identifiable

Let $X_1$,...,$X_m$ be i.i.d. F, $Y_1$,...,$Y_n$ be i.i.d. G, where model {(F,G)} is described by $\hspace{20mm}$ $\psi$($X_1$) = $Z_1$, $\psi$($Y_1$)=$Z'_1$ + $\Delta$, where $\psi$ is an unknown ...
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Likelihood Functon.

$n$ random variables or a random sample of size $n$ $\quad X_1,X_2,\ldots,X_n$ assume a particular value $\quad x_1,x_2,\ldots,x_n$ . What does it mean? The set $\quad x_1,x_2,\ldots,x_n$ ...
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Likelihood Function for the Uniform Density $(\theta, \theta+1)$

Let the random variable X have a uniform density given by $f(x;\theta)$~$R(\theta,\theta+1)$ What is the maximum likelihood function according to the samples $X_1,\ldots,X_n$? The question is much ...
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101 views

Distribution of likelihood ratio in a test on the unknown variance of a normal sample

EDIT: I have followed up to this discussion with a second question: Hypothesis test on variance of normal sample I am preparing for a stat exam and I was trying to derive the distribution of the ...
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72 views

How to compute the unique MLE from an Exponential Family of Distributions?

Let $$ f(x;\theta)=\frac{1}{\pi} \frac{e^{\theta x}\cos(\theta \pi/2)}{\cosh(x)}, x\in{\mathbb{R}} $$ be a family of densities and which is clearly exponential family. Then what is the Maximum ...
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Show that as $d$ goes to $\infty$, a standardized version of $X$ has the STD Normal Dist

I am currently stuck on this problem and I would greatly appreciate some help. The problem is as follows: Let $X$ have a chi-square with $d$ degrees of freedom. Show that a standardized version of ...
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1k views

Sufficient Statistic for a Geometric R.V.

I have a problem that I know I am very close to the solution for, but I think I just need some more formatting to make it a really clean proof. The problem goes like this: Suppose X is a discrete ...
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Iterative Mean, Covariance Algorithm Convergence

The problem is to show that the following iterations converge to the vector $\mu$ and the matrix $\Sigma$. We have data in the form of nx1 vectors $\mathbf{Q}_k$, $1 \leq k \leq N$ where ...
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Why is there a difference between a population variance and a sample variance

Sorry if this answer is simple but I was wondering why is there a difference between a population variance and a sample variance? I understand The variance is calculated as: $$\text{Var} = ...
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statistics inequality

Let $\theta$ be a discrete pararmeter and $\gamma_{n}$ be an estimator. Prove that for any $c>0$ we have that $$\text{E}[(\gamma_n-\theta)^2] \ge\Pr[|\gamma_n-\theta|>c]\cdot c^2$$
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193 views

How to calculate probability using multinomial distribution?

So according to the multinomial distribution, the probability function $\Pr(X_1 = x_1, X_2 = x_2, \dots, X_k = x_k)$ is equal to $\dfrac{n!}{x_1! x_2! \cdots x_k!} \cdot p_1^{x_1}\cdot p_2^{x_2} ...
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Two definitions of Bayes Sufficiency

"Bayes Sufficiency" is defined in two ways. Are they equivalent? Setting A statistical experiment $S$ is a triplet $\left(\left(\Theta,\mathcal{F}\right),\left(\Omega,\mathcal{A}\right),P\right)$, ...
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607 views

Rao-Blackwell Uniform Distribution

I am having a bit of an argument with my study group about a Rao-Blackwell problem that we have for our statistical theory class. The problem goes like this: Let X~U(0,$\theta$), and suppose we have ...
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Application of Multivariate Analysis

The following situation is proven valuable where multivariate analysis can be applied. This example is taken from the book Applied Multivariate Statistical Analysis ...
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126 views

Bayes estimator from a geometric distribution with a uniform prior

X is a random variable with Ber(p), 0 Y is the number of trials until a success occurs. Assume the prior p is unif(0,1). I have trouble in figuring out the posterior density f(p|Y). With the ...
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Correct choice of analytical statistic method for time series problem

I want to make a statement about corruption (y) influenced by the ratification of the UN-convention-Against-Corruption "contract" (x). Luckily, most of them signed it within 3 years. So, I have ...
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112 views

Maximum likelihood estimator of $P(X < y)$ for fixed $y$

I'm having a problem understanding the following question. Given the following density function $f_X(x; \theta) = (\theta +1)x^\theta$ on $0<x<1$, find the maximum likelihood estimator for ...
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If $P$ is the set of all distributions, the only sufficient subfield is the trivial one

According to an article by Bahadur, if $P=\left\{p\right\}$ is the set of all probability measures on the measurable space $\left(\Omega,\mathcal{A}\right)$, $\mathcal{A}$ is the only possible ...