The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.

learn more… | top users | synonyms

1
vote
1answer
1k views

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 ...
0
votes
1answer
63 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 $\sigma = 0.6$ ...
0
votes
1answer
781 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} = \dfrac{\sum_{i=...
3
votes
2answers
136 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 ...
2
votes
2answers
101 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 ...
2
votes
2answers
47 views

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 $...
1
vote
1answer
428 views

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 ...
1
vote
1answer
448 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 ...
0
votes
1answer
90 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 ...
-2
votes
1answer
55 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\...
-2
votes
1answer
30 views

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 ...
8
votes
1answer
10k views

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 ...
0
votes
2answers
156 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 ...
2
votes
1answer
162 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 ...
2
votes
2answers
148 views

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 ...
2
votes
1answer
120 views

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 $\quad x_1,x_2,\ldots,x_n$ ...
6
votes
1answer
2k 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!
4
votes
2answers
61 views

I am running a series of experiments that I expect to have similar outcomes. What is the best method to measure statistical significance?

Following on from this comment on an answer to my previous question, I'd like to know two things: what the best statistical test I can use to measure significance on the experiments I'm running? (...
3
votes
2answers
72 views

Reasoning for confidence interval

Suppose $$X_1,\dots,X_{20} \sim f_X(x;\beta)$$ where $$f_X(x;\beta) = \frac{1}{\beta} e^{-\frac{x}{\beta}},\quad x>0;\beta>0$$ It can shown that ("details omitted") $$P(0.52 \bar{X} \leq \beta ...
3
votes
2answers
110 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}{...
2
votes
2answers
63 views

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 $X$...
2
votes
1answer
41 views

why equal variance assumption is necessary in T-test

So I generally understand the basis for the t-test: i.e. you take advantage of the fact that you can make $\bar{X}-\bar{Y}$ standard normal: $$Z = \frac{((\bar{X}-\bar{Y})-(\mu_X-\mu_Y))}{\sigma \sqrt{...
2
votes
1answer
81 views

Which test should I use for hypothesis testing with a small sample size?

I've run a test with one control and one experiment group, and am questioning myself on whether or not I've used the right test (or if significance can even be calculated on the following sample sizes)...
1
vote
1answer
66 views

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 ...
1
vote
1answer
163 views

Two definitions of Bayes Sufficiency

"Bayes Sufficiency" is defined in two ways. Are they equivalent? Setting A statistical experiment $S$ is a triplet $\left(\left(\Theta,\mathcal{F}\right),\left(\Omega,\mathcal{A}\right),P\right)$, ...
1
vote
1answer
37 views

In OLS is the vector of residuals always 0? [duplicate]

I am trying to show that $$\sum_{i=1}^ne_i = 0$$ 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 solution is as follows: $$e'1 = Y'(I-...
1
vote
0answers
17 views

Relation between Bayesian analysis and Bayesian hierarchical analysis?

I have been studying a Bayesian hierarchical model. In that model all I am dealing is with the estimation of parameters. In Bayesian analysis, loosely speaking, we update our prior knowledge (in light ...
1
vote
1answer
122 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 $P(X&...
1
vote
1answer
362 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} \...
1
vote
1answer
38 views

Correctly calculating the bias of an estimator

I'm currently learning about method of moments and maximum-likelihood estimators and have confused myself with this issue: First, let me estimate the parameter $\lambda$ from the exponential ...
1
vote
0answers
174 views

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 $$\begin{...
1
vote
1answer
1k 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 ...
1
vote
1answer
96 views

statistics inequality

Let $\theta$ be a discrete pararmeter and $\gamma_{n}$ be an estimator. Prove that for any $c>0$ we have that $$\text{E}[(\gamma_n-\theta)^2] \ge\Pr[|\gamma_n-\theta|>c]\cdot c^2$$
0
votes
1answer
29 views

Determining the degree of freedom for a $\chi$-squared test

I have read that the degree of freedom is calculated by subtracting $1$ from the number of states a random variable can be in. I am performing a goodness of fit test on a $64\times 32$ matrix where ...
0
votes
1answer
40 views

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 ...
0
votes
1answer
62 views

Defining bias function for n trial

Let a point estimate for the sample variance be given as $\hat{\sigma}^2 = \frac{1}{n}\sum\limits_{i=1}^n(X_i- \bar{X})^2$ where $n$ is the number of samples. What is the bias in this estimate as a ...
0
votes
1answer
50 views

Proving that a statistic is not sufficient (uniform case).

Let $X=(X_1,...,X_n)$ be i.i.d. $U(0,\theta)$. How to show that $$\frac{2}{n}\sum_{i=1}^{n}X_i$$ is not a sufficient statistic? I have already proven that $\max_{i=1,...,n}X_i$ is a sufficient ...
0
votes
1answer
42 views

Estimating unknown weights of various parameters in an equation

I have an equation which has some unknown weights attached to various parameters. None of the weights are known. However, I have a history of data available with me which can be used to predict the ...