Questions tagged [statistical-inference]

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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Distribution of Sum of Discrete Uniform Random Variables

I just had a quick question that I hope someone can answer. Does anyone know what the distribution of the sum of discrete uniform random variables is? Is it a normal distribution? Thanks!
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The derivation of the Wald interval

I'm asking about the binomial proportion confidence interval, also known as the Wald interval. Recall that $$\lim_{n \to \infty}{P_p \left( -z_{1-\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}}+\bar{X_n} \...
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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} = \dfrac{\sum_{i=...
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Spectral densities of finite dimensional sample covariance matrices

Let $N \ge 2$ and $T > N$ be integers. In multivariate statistics it is of interest to analyze spectra of sample covariance matrices. The resolvent ${\mathfrak g}_M(z)$ encapsulates the whole ...
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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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1answer
973 views

Justifying the Normal Approx to the Binomial Distribution through MGFs

Would absolutely love if someone could help me with this question, in a step by step way to help those who are uninitiated to Statistics and Mathematics. So, I am trying to "prove/justify" through ...
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149 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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Maximum Likelihood Estimator of parameters of multinomial distribution

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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Probability vs Confidence

My notes on confidence give this question: An investigator is interested in the amount of time internet users spend watching TV a week. He assumes $\sigma = 3.5$ hours and samples $n=50$ users and ...
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How to determine the number of coin tosses to identify one biased coin from another?

If coin $X$ and coin $Y$ are biased, and have the probability of turning up heads at $p$ and $q$ respectively, then given one of these coins at random, how many times must coin A be flipped in order ...
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843 views

Finding an unbiased estimator of $e^{-2\lambda}$ for Poisson distribution

If $X_1,X_2,\ldots,X_n\sim \mathrm{Pois}(\lambda)$, find an unbiased estimator of $e^{-2\lambda}$. I am actually supposed to find the UMVUE of $e^{-2\lambda}$ but I first have to find its unbiased ...
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Minimal sufficient statistic of $\operatorname{Uniform}(-\theta,\theta)$

I am seeking clarification on why both the vector $(X_{(1)},X_{(n)})^T$ and $\max\{-X_{(1)},X_{(n)}\}$ are sufficient for $\operatorname{Unif}(-\theta,\theta)$, but only $\max\{-X_{(1)},X_{(n)}\}$ is ...
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Fisher Information for Geometric Distribution

Find the Cramer-Rao lower bound for unbiased estimators of $\theta$, and then given the approximate distribution of $\hat{\theta}$ as $n$ gets large. This is for a geometric($\theta$) distribution. ...
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298 views

Poisson random variables and Binomial Theorem

I'm working on a problem from Casella and Berger's Statistical Inference. X is distributed as Poisson$(\theta)$ and Y is distributed as Poisson$(\lambda)$, with X and Y being independent. We let U = X ...
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Cramer Rao lower bound in Cauchy distribution

I need to calculate the Cramer Rao lower bound of variance for the parameter $\theta$ of the distribution $$f(x)=\frac{1}{\pi(1+(x-\theta)^2)}$$ How do I proceed I have calculated $$4 E\frac{(X-\...
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Maximum Likelihood Estimate with different parameters

Suppose that X and Y are independent Poisson distributed values with means $\theta$ and $2\theta$, respectively. Consider the combined estimator of $\theta$ $$ \tilde{\theta} = k_1 X + k_2 Y $$ where $...
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Prove $(\sum_{i=1}^{n}X_{i},\sum_{i=1}^{n}X_{i}^{2})$ is not a complete statistic for $N(\mu,\mu^2)$ distribution

Let $X_{1},\ldots,X_{n}\stackrel{\text{ i.i.d }}{\sim}N(\mu,\mu^{2})$. $T=\left(\sum_{i=1}^{n}X_{i},\sum_{i=1}^{n}X_{i}^{2}\right)$ is a sufficient statistic for $\mu$. Also $T$ is minimal sufficient....
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1answer
18k views

Shifted Exponential Distribution and MLE

I was doing my homework and the following problem came up! We have the CDF of an exponential distribution that is shifted $L$ units where $L>0$ and $x>=L$. The CDF is: $$1-e^{-\lambda(x-L)}$$ ...
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Let Y(1), Y(2), Y(3), Y(4), Y(5) denote the order statistics of a random sample of size 5 from a distribution having p.d.f.

Help me to solve this problem please.. Let $Y_{(1)}, Y_{(2)}, Y_{(3)}, Y_{(4)}, Y_{(5)}$ denote the order statistics of a random sample of size 5 from a distribution having p.d.f. $f(y) = e^{(-y)}, 0 ...
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566 views

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} = \frac{1}{...
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Checking the consistency and Bias of $\frac{\sum X_i +\sqrt{n}/2}{n+\sqrt{n}}$

Let $X_1,\ldots,X_n$ be i.i.d. $B(1,\theta)$ random variables, $0<\theta<1$. Then, as an estimator $\theta$, check if $T(X_1,\ldots,X_n)= \dfrac{\sum_{i=1}^n X_i +\sqrt{n}/2}{n+\sqrt{n}}$ is ...
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1answer
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Bayesian networks problem

Attempting to understand Exercise 20 (pdf page 44) in this book: The party animal problem corresponds to the network in g(3.14). The boss is angry and the worker has a headache - what is the ...
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In Bayesian Statistic how do you usually find out what is the distribution of the unknown?

To estimate the posterior we have $$p(\theta|x) = \frac{p(\theta)*p(x|\theta)}{\sum p(\theta ')*p(x|\theta ')}$$ $x$ is usually the experimentally sampled data, and $\theta$ is the model, but both $...
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UMVUE for $\theta^2$

Let $X_1,...X_n$ be a random sample with distribution $\text{Normal}(\theta,1)$. Find the UMVUE for $\theta^2$ What I´ve done so far: I have already shown that $T=\sum_{i=1}^nX_i$ is a complete ...
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2answers
148 views

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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1answer
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How do I show that the sum of residuals of OLS are always zero using matrices

I am trying to show that $$\sum_{i=1}^ne_i = 0$$ using matrices (or vectors). I have two hints, so to speak: $$ HX = X$$ where $H$ is the hat matrix, and that $$\sum_{i=1}^ne_i = e'1$$ My previous ...
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1answer
640 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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What hypothesis test is suitable in this case?

Null: Event frequency does not vary by weekday Alternate: Event frequency varies by weekday Data: Day: Mon, Tue, Wed, Thu, Fri, Sat, Sun event_count: 12, 15, 20, 10, 19, 10, 11 What hypothesis test ...
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134 views

solve the integral equation 2

I want to solve the integral .Solve is difficult. I want to use statistical methods to solve them. $$\int_{0}^{+\infty}x \exp\{ ax-b x^2\}d x=\int _{0}^{+\infty} x\exp\{-b(x^2-\frac{a}{b}x)\}dx=\\ exp\...
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1answer
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Identifying joint distribution

Let $Y_1$ and $Y_2$ be independent random variables with $Y_1\sim N(1,3)$ and $Y_2 \sim N(2,5).$ If $W_1=Y_1+2Y_2$ and $W_2=4Y_1-Y_2$ what is the joint distribution of $W_1$ and $W_2$? Is my ...
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Why the sum of residuals equals 0 when we do a sample regression by OLS?

That's my question, I have looking round online and people post a formula by they don't explain the formula. Could anyone please give me a hand with that ? cheers
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When does the variance of a consistent estimator go to zero?

I came across the following statement (marked as true) in multiple-choice section of an old exam: The variance of a consistent estimator goes to zero with the growing sample size. As far as I can ...
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How to find the maximum likelihood estimators of parameters in the Pareto distribution? [closed]

Here's the Pareto distribution: $$F(x; \theta_1, \theta_2) = 1 - \Big(\frac{\theta_1 }{x}\Big)^{\theta_2}, \qquad \theta_1 \le x, \qquad \theta_1, \theta_2 > 0$$ I have been trying to solve the ...
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2answers
735 views

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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1answer
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Bayesian Approach: Is a die from a 3-D printer fair?

In a recent post "Fair die or not from 3-D printer"on this site @Eumel reported making a die on a 3-D printer, providing data on the faces seen in 150 rolls, and wondered about "the chances that the ...
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2answers
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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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1answer
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Is there any meaning of this “Median-mean”

Given a data set $\{a_1,\cdots,a_n\}$ with median M Define the medimean to be the value of $x$ s.t. $$\left(\frac{1}{n}\sum_{n}{}{a_n}^x \right)^{1/x}=M$$ Is this $x$ value useful / used at all in ...
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Reference request, statistical inference

Good morning, I'm looking for a good reference for study on statistical inference, the main topics that will study are Tests of Hypotheses Interval estimation I recommend taking a look at Mood ...
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How to find UMVUE of $\theta^k$ with bernoulli distribution.

Let $x_1, x_2, \cdots, x_n$ be a random sample from the Bernoulli ($\theta$). The question is to find the UMVUE of $\theta^k$. I know the $\sum_1^nx_i$ is the complete sufficient statistics for $\...
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1answer
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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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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 $ P(...
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2answers
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Derive an unbiased estimator for $\theta$.

Exercise: Let $X_1,\ldots,X_n$ be a random sample from the distribution with density $$f(x|\theta) = \dfrac{2x}{\theta^2}\mathbb{1}_{(0,\theta)}(x)$$ w.r.t. the Lebesgue measure. Derive an unbiased ...
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B-Splines and sum of uniform variables

Exercise 5.2 in Elements of Statistical Learning Goal is to show that an order $M$ B-Spline basis function is the density function of a convolution of $M$ uniform random variables. Although I feel ...
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2answers
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Show/Prove that $F_{\alpha,n,m} =1/F_{1-\alpha,m,n}$

Show/Prove that $F_{\alpha,n,m} = \frac{1}{F_{1-\alpha,m,n}}$ The distributions I'm working with are the Fisher distribution and the Chi-square distribution. I can prove that n and m switch for the F ...
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1answer
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Minimal sufficient statistics for uniform distribution on $(-\theta, \theta)$

Let $X_1,\dots,X_n$ be a sample from uniform distribution on $(-\theta,\theta)$ with parameter $\theta>0$. It is easy to show that $T(X) = (X_{(1)},X_{(n)})$ is a sufficient statistic for $\theta$ ...
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2answers
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The Wald test with Poisson distribution

Let $X_1,\ldots, X_n\sim \operatorname{Poisson}(\lambda)$. Let $\lambda_w>0$ be given, I am trying to find the size $\alpha$ Wald test for $H_0$: $\lambda=\lambda_w$ vs $H_1$: $\lambda\neq \...
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1answer
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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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1answer
705 views

Prove that the the variance estimator $\widehat{\sigma}^2=MSE/(n-2)$ is biased is the simple linear regression model

This is in scope of the simple linear model. Im trying to prove that $\mathbb{E}\left(\widehat{\sigma}^2\right) = \sigma^2$ for $$\widehat{\sigma}^2 = \frac{1}{n-2}\sum^n_{i=1} \left(y_i-\widehat{y}...
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1answer
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Conditional expectation $E(X_1 \mid \overline{X}_n)$ if $X_1,\dots,X_n$ are i.i.d. Am I correct?

Conditional expectation $E(X_1 \mid \overline{X}_n)$ if $X_1,\dots,X_n$ are i.i.d. Since $X_1,\dots,X_n$ are i.i.d, then $E(X_1 \mid \overline{X}_n) = E(X_1)=\overline{X}_n$ Am I correct in thinking ...
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3answers
1k views

Proof that $\frac{(\bar X-\mu)}{\sigma}$ and $\sum_{i=1}^n\frac{(X_i-\bar X)^2}{\sigma^2}$ are independent

Let $X_i\sim N(\mu,\sigma^2)$ ; where$[i=1,2,\ldots,n]$ $Z_i\sim N(0,1)$ ; where$[i=1,2,\ldots,n]$ Proof that $\bar Z=\frac{(\bar X-\mu)}{\sigma}$ and $\sum_{i=1}^{n}(Z_i-\bar Z)^2=\sum_{i=1}^n\frac{...