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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interpretation of gross reproduction rate and net reproduction rate

If GRR=1.234 then what can be said about the population growth? And then what can be said when net reproduction rate, NRR=1.234?
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Variance- covariance matrix

Consider $H$ denotes hat matrix and $e$ denotes residual. In the book Applied regression Analysis by Draper/Smith, it is written that : $\mathbb V(e_i)$ is given ...
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206 views

Finding UMVUE from Lehmann-Sheffe Directly

I am having some trouble with an example from the book I am following. Let $X_1,X_2,...,X_n$ for $n>2$ be an iid set of $N(\mu,\sigma^2)$ random variables with, $\mu\in\mathbb{R}$ and ...
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Is difference between two sets of measurements significant?

Consider the following experimental setting: I have two machines $m_0$ and $m_1$ of which I would like to know which one performs better. For this I have set up an experiment to measure the time it ...
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Inferential statistics question [on hold]

The consumption of sugar is 63 lbs per year. A sample of 16 adults found a mean of 60 lbs per year with a standard deviation of 20 lbs. At a 90% confidence level can we support the 63 lbs population ...
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Question about Logistic Regression - 2

How should I tell the difference between those two formulas in the circles below. I am studying logistic regression and I have faced two different formulas from two different documents. I don't know ...
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17 views

Biased MLE estimate of mean (expectation)

Please give an example of p.m.f. or p.d.f. , the maximum likely-hood estimate of whose mean (expectation) is a biased estimator . Thanks
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223 views

One-tailed two-sample T-test OK?

I'm trying to conduct a one-sided hypothesis test between two random variables which are both asymptotically normally distributed with different variances. The variances are not known but have been ...
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22 views

Confidence Interval for Regression Coefficient ,$\beta$

In the book 'Applied regression Analysis' by Draper/Smith, it is written that : Obtain individual $100(1-\alpha)\%$ confidence interval for the various parameters separately from the formula ...
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Calculating optimum values of $u$ and $m$ from $\mathbb V(\bar {y_2}\prime)=\frac{S_2^2(n-u\rho^2)}{n^2-u^2\rho^2}$

I have to find optimum sample size in sampling on two occasions. Suppose that the samples are of the same size n on both occasions. In selecting the second sample, $m$ of the units in the first ...
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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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22 views

How do I convert non-normal distribution to a normal distribution?

I have the below graph I drew using histogram in Excel but when I got my data and graphed it, certainly it is not a normal distribution. My assignment required me to gather some data and draw normal ...
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Mantel-Haenszel $\chi_1^2$ statistic

I was doing a particular example from the book Epidemiologic Research by Kleinbaum(example 15.6) and didn't understood some basic statistical aspect. ...
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Is it true that $\mathbb E[{\frac{X}{Y}]}={\frac{\mathbb E[X]}{\mathbb E[Y]}}$?

If $X$ and $Y$ are both random variables, does it hold $$\mathbb E\left[\frac{X}{Y}\right]={\frac{\mathbb E[X]}{\mathbb E[Y]}}$$ ??
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19 views

Statistics on a column of values

I have a class project that I am working on as an engineering student but the course is probability and statistics so I don't really know how to make sense or use of what I am learning so I thought of ...
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1answer
3k views

How to calculate the covariance matrix

I tried searching a lot on the net and got the following sources: Source One Source Two The first source seems to be incorrect cause when I calculate it using matlab it comes to be different from ...
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34 views

Method of moments for Beta $(\alpha_1,\alpha_2)$ distribution

I am trying to solve for the first two moments of a Beta$(\alpha_1,\alpha_2)$ distribution. We know that the first moment is equal to: $\mu_1 = \frac{\alpha_1}{\alpha_1+\alpha_2}$ and the second ...
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1answer
22 views

Confidence Intervals with two variables

A random sample of $50$ men shows the following relation between annual income $Y$ (in dollars) and education $X$ (in years): estimator for $Y, (\hat{Y}) = 1200 + 800X $ Average income is $Y = ...
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Alternatives to absolute error?

Let me explain my scenario in which I need to calculate absolute error. Lets say the X is the actual value. And X' is the value of X with some error 'e'. So X' = X + e'. Lets say i = 1 to 10000. I ...
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“Approximation” of a maximum likelihood confidence set

I have some trouble constructing the "approximate" set, how can it be defined or calculated? Suppose that three characteristics in a large population can be observed according to the following ...
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A question about $\chi^2$ distribution

Ok, i have a question but i start with a definition first so that one can get the context. (All variables in question have the same variance and under $H_0$ which we are considering - they have the ...
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What is sampling with unequal probabilities without replacement

sampling without replacement: i know if my sample size is $n$, and if $i^{th}$ unit is drawn then we will not replace the unit in the population , so that in my next draw i can get any unit except ...
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Can I use a z-transformation instead of Welch-t-test

I am not a big mathematician and at the moment I am just refreshing some of my statistical knowledge. So, when doing a students t-test with two independent samples, a precondition is, that both ...
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384 views

Alternative Hypotheses with equal sign

I have been having some trouble formalizing the hypotheses for an experiment. My claim, it's not this one, but it basically poses the same problem I am having: Studying math in the morning or in ...
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UMVUE using complete and sufficient statistic

Let $X_1,X_2,...,X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. I showed that $(\bar X,S^2)$ is jointly sufficient for estimating ($\mu$,$\sigma^2$) where ...
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Maximum Likelihood estimators in linear models

Consider two simple linear models. $y_{1j}=\alpha _1+\beta_{1}x_{1j}+\epsilon_{1j}$ and $y_{2j}=\alpha _2+\beta_{2}x_{2j}+\epsilon_{2j}$ , $ j=1,2,...,n>2$ where $ ...
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Finding UMVUE for Poisson distribution using Rao Blackwell

Let ${X_1,X_2,... ,X_n}$ be a random sample from a Poisson distribution with parameter $\lambda$.Let $\gamma(\lambda)=P(X<=1)$. Find UMVUE for $\gamma(\lambda)$. This is my attempt: First I ...
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1k views

What situation calls for dividing the standard deviation by $\sqrt n$?

While doing my homework and checking my answers with the book's answers I noticed that sometimes the standard deviation is divided by $\sqrt n$ where $n$ is the sample size. I'm a little confused. For ...
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25 views

Statistical inference

Ten balls are picked without replacement from a box of 50 balls. All turn out to be red. A argues that the box contains only red balls. B argues that it contains 25 balls each of red and and some ...
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Draw and compare the likelihood using R

The following shows the heart rate (in beats/minute) of a person, measured throughout the day: 73, 75, 84, 76, 93, 79, 85, 80, 76, 78, 80. Assume the data are an iid sample from ...
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What is the problem with this model parameter estimation algorithm?

In a statistical model with parameters $\theta$ and unobserved laten variables $Z$, the model likelihood is $$L(\theta;X)=Pr(X|\theta)=\sum_ZPr(X,Z|\theta)$$ The standard way to estimate $\theta$ ...
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Method for detecting anomalies in timeseries?

I am exploring creating alerts for anomalies in a timeseries. Say that I have a graph like this: There are two anomalies here that I would wanted to be alerted on: First, on April 7th with there ...
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35 views

Power testing and a Poisson Distribution

Let $X_1, ..., X_n$ be a random sample from a Poisson distribution with parameter $\lambda$. If $X$ is Poisson with parameter $\lambda$, then $\text{E}[X] = \text{Var}[X] = \lambda$; hence ...
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Maximum Likelihood Estimator for a Poisson random variable

$X$ is distributed as a Poisson random variable with parameter $\lambda$ if it has the probability mass function: $f(x; \lambda) = \dfrac{e^{-\lambda}\lambda^x}{x!},\ x = 0,1,2,\ldots$ Find the ...
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Confidence Intervals and Inferences

I need help with c, as I have attempted a and b already, but believe they help with context. Suppose you took a random sample of 100 accounts in a large department-store chain, and found that the ...
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Population Proportions

50% of teens believe that we need to address the problem of climate change. If a random poll of 1500 teens was taken (about the size of the Gallup poll), what is the chance that its proportion would ...
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Numerical CDF estimation for complicated random variable

Given a combination $U$ of several random variables $X,Y,Z...$ with known distributions, what is an efficient numerical algorithm to estimate PDF or CDF of $U$, if its CDF, PDF, characteristic ...
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Parameter estimation of Lorenz system (nonlinear dynamical system)

My problem is as follows. I have to estimate parameters of Lorenz system using given data. Lorenz system is described by following system of ODEs: $$ \frac{dx}{dt} = \sigma(x-y) \\ \frac{dy}{dt} = ...
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Sampling from a graph

Suppose you have a graph $G=(V,E)$ that is unobservable globally and you wish to take a sample from the vertices of that graph to infer something about its global properties from local properties. ...
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Model of three mutually exclusive events to be accepted or rejected with confidence .05

According to a genetic model, three mutually exclusive characteristics A, B, C are present with probabilities: P(A) = 0.1, P(B) = 0.2, P(C) = 0.7 In a random sample of size 100, A happens to appear ...
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Show there is no UMP test for $N(\mu,\sigma^2)$

Here's a testing problem. $X_1, \cdots, X_n, indep, \sim N(\mu, \sigma^2)$ where $\mu, \sigma^2$ are unknown. $H_0: \mu \le \mu_0$ vs $H_1: \mu > \mu_0$ where $\mu_0$ is specified. I ...
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Matrix Completion feature vectors

In the noisy matrix completion problem, with enough number of revealed entries say $|E| = nr^2 log(n) $, can we have a bound on the error in the singular vectors of a sub matrix. For, example say ...
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454 views

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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learning phenomena in bayesian technique

I have identified using bayesian method for my auto tagging application. I am dealing with user facebook post. Post belongs to ...
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31 views

Question about Logistic Regression - 7

I am currently studying Logistic Regression. I am facing a problem with understand the sentence in the red circle below. I am trying to figure out what he/she means by the sentence. Please let me have ...
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23 views

Question about Logistic Regression - 6

I am studying Logistic Regression and I have come across to understating the paragraph below. I kind of can understand, but it makes me confused when I read the sentence in the red circle, "It also ...
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Question about Logistic Regression - 5

Can somebody give me a clear explanation about logistic regression in bold below? Logistic regression can be binomial or multinomial. Binomial or binary logistic regression deals with situations in ...
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Question about Logistic Regression - 4

I am currently studying on logistic regression. So I have found a document on the Internet explaining about it. Somehow, it explains Bernoulli distribution in the beginning and I am having a problem ...
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Question about Logistic regression-1

As I am currently studying Logistic regression, but I am still new to this. I have encountered a sentence and a word below in red circles that I can only image but not with my own clear ...
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Whether or not $X_1$,…,$X_n$ are independent and exchangeable

For some n = 1,2,..., let $Y_1$,...,$Y_{n+1}$ denote iid real-valued random variables. Define $X_j$ = $Y_j$$Y_{j+1}$, $\hspace{10mm}$j=1,...,n a) Are $X_1$,$X_2$,...,$X_n$ independent? b) Are ...