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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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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26 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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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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15 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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29 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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24 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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41 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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24 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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33 views

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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14 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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28 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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49 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 ...
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1answer
34 views

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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13 views

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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35 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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27 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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17 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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50 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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7 views

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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12 views

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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42 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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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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Find $Var(M_2)$ when $X_i \sim N(\mu,\sigma^2)$.

Find $Var(M_2)$ when $X_i \sim N(\mu,\sigma^2).$ $M_2=(\Sigma^n_{i=1}{(X_i-\bar{x})}^2)/n$ Then, is it mean that the question is ask to find $S^2_n$ ? $Var(S^2_n) = n^2/(n-1)^2Var(M_2)$ $= ...
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45 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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40 views

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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1answer
18 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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12 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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103 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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10 views

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 ...
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1answer
40 views

Probability density function transformation

Probability density function of f is given as a uniform distribution, f(x)=1 and I need to find the probability distribution function of Y=X-X^2. What I have done so far is that I found the inverse ...
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1answer
22 views

Compute joint density function of exponential fuction

Consider a set of continuous random variablces $Y_1 ... Y_n$, i.i.d, exponentially distributed . with rate parameter $\lambda$. I showed first that for one single variablce (ie the first) its ...
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2answers
34 views

Discrete mathematics vs. non parametric statistics

Is there any meaningful connection betveen non parametric statistics and discrete mathematics? I am reading this book: ...
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32 views

Comparing models to smoothed data

I am attempting to fit a model to a noisy data set. I am performing this modeling in two stages - first, smoothing it out by fitting an analytic mixture model to it, and second, fitting my final model ...
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27 views

how to prove independence in this case

The question is : $X_1,...,X_n$ are i.i.d.$Uniform(0,\theta)$. Let $X_{(n)}$ denote the maximum of these $n$ random variables. Prove that $\frac{X_1}{X_{(n)}}$ and $X_{(n)}$ are independent. What I ...
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43 views

Joint Probability of Random Variables

Suppose I took measurements $\{X_i\}$, which are all independent and they follow a normal distribution $X_i\sim N(\mu,\sigma)$. I am asked for the joint probability of all of the measurements. Based ...
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1answer
32 views

Exchangeable/Independent Bernoulli Distribution

Let P be a uniform random variable on the interval $(0,1)$ with density function f(p) = 1, $0<p<1$. Let $X_i|P$, i = 1,2,...,n be independent and identically distributed random variables having ...
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1answer
33 views

Maximum Likelihood Question

The aim is to find the maximum likelihood estimator for theta. $f(x)$ is given and we can assume that $1\le x\le-1$. I have completed the steps seen in the image, however I am having difficulty ...
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Maximum Likelihood Estimation Question

I'm really struggling with this question. From my understanding in order to find the maximum likelihood estimator for theta, the function needs to be partially differentiated with respect to theta ...
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1answer
20 views

Central Limit Theorem & Delta method problem

Let $U_1$,...,$U_n$ be a random sample from the U(0,1) a. Let $X$=-log($U$). Find the distribution of X b. Let $Y$=$1/{\prod_{i=1}^n U_i^{1/n}}$, where $U_1$,...,$U_n$ be a random sample from the ...
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Statistics / Probability / Population Proportion / Probability of Successive Positive Outcomes

I am not really sure where to start. "While researching fish populations in a specific lake in Arkansas, it is noticed that only 1 out of 9 bass is caught after another bass (implying no other types ...
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48 views

Show expectation is infinite

Let $X_1,\ldots,X_n$ be independent, identically distributed with expectation 1 and finite variance. Find the limit distribution of $\sqrt{n}(\bar{X}_n^{-1}-1)$. If the random variables are sampled ...
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Qual problem: One dim. sufficient statistic for $\lambda$ based on the data (X,Y) where X-Piosson$(\Lambda)$ and Y-Bernolli$(\lambda/(1+\lambda))$

Qual problem: We observe the pair $(X,Y)$ where $X$-Poisson$(\lambda)$ and $Y$-Bernoulli$(\lambda/(1+\lambda)),$ $\lambda$ is unknown. Find one dimensional sufficient statistic for $\lambda$ based ...
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39 views

Difficult Survey Sampling question

Question: A Secretary of State wants to survey the primary owners of motorcycles registered in the state to estimate the proportion who want the license plates redesigned. (Primary owner means that ...
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15 views

Show that the found value is the MLE

Let $ X_1, ... X_n$ i.i.d with pdf $$f(x;\theta)=\frac{x+1}{\theta(\theta+1)}\exp(-x/\theta), x>0, \theta >0$$ It is asked to find the MLE estimator for $\theta.$ The likelihood function is ...
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2answers
60 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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1answer
23 views

Showing something converges, in distribution, to a normal distribution

I'm not sure how relevant the first few parts are, but I will post it just in case... $(X_i,Y_i), i=1,\dots,n$ are independent where $X_i$ has an exponential distribution $\mathcal{E}(\lambda_i)$ ...