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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how to find relationships in dataset with multiple variables

I have a large project data set ,which includes numeric values like dollar amounts, and non numeric quantities like country codes, purpose codes etc I want to find relationships between the variables. ...
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8 views

interpreting the multivariate Kalman filter update equations

consider a multi-dimensional Kalman filter model with these state transition and measurement probabilities: $P(x_{t+1} | x_{t}) = Normal(Fx_{t}, \Sigma_{x})$ $P(z_{t} | x_{t}) = Normal(Hx_{t}, ...
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1answer
13 views

Do we need to check that maximum likelihood estimator is a maximum?

For maximum likelihood estimation, do we theoretically need to check that the critical point is a maximum (rather than a minimum or saddle point) or is this automatic? I believe that it is automatic ...
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9 views

Showing that moment estimates are asymptotically bi-variate normal.

Let $X_1,\dots,X_n$ be iid $\Gamma(p,1/\lambda)$ with density $g_\theta (x) = \frac{1}{\Gamma(p)} \lambda^p x^{p-1} e^{-\lambda x}$, $x>0$, $\theta = (p,\lambda)$, $p > 0$, $\lambda > 0$. ...
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1answer
19 views

Geometric distribution $G(p)$ for independent random variables $X$ and $Y$

Question: If the random variables $X$ and $Y$ are independent and each have the geometric distribution $G(p)$ - that is, $P(X=k)=P(Y=k)=pq^k$ for $k=0,1,2,\ldots$ (where $q=(1-p)$) show that: (I) ...
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4 views

Transformation of the LR-test for normal data

Given an iid normal sample $X_1,...,X_n$ with unknown mean $\mu$ and unknown variance $\sigma^2$, we want to test $H_0:\sigma^2=\sigma_0^2$ vs. $H_1:\sigma^2\not=\sigma_0^2$ using the likelihood-ratio ...
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29 views

Function Looks Poisson-Like: But What's the Parameter $\lambda$?

(On pause) I have $$f\left(x\right)=-x\left( x\sqrt{4-x^2}-4\arccos\left(\frac{x}{2}\right) \right)\arccos\left(\frac{x^2+d^2-1}{2dx}\right)$$ which looks a bit like the continuous version of ...
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25 views

Bivariate density function

For a distribution function $F$ density function is defined as $\dfrac{\partial^2 F}{\partial x \,\partial y}$. Is it essential that $F$ is differentiable? Is it required that $\dfrac{\partial^2 ...
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16 views

Hypothesis testing method for a single measurement variable but three nominal variables

I'm finding it difficult to get a hard and true answer to which test I should use to determine whether the differences in my data are significant. In short: I have 1 measurement variable and 3 ...
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2answers
28 views

Fallacy of denying the hypothesis

I need help with this question Im not sure what a fallacy of denying the hypothesis is. Use a truth table to show that $p\to q$ and $\neg p$, $\therefore \neg q$ is not a valid rule of inference. It ...
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1answer
30 views

Power of a test [closed]

Can someone explain me how to solve this, with details? A seed producer claims that at least 90% of the seeds germinate. The producer’s claim is tested with a significance level of 1%. a) Determine ...
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1answer
24 views

What is the meaning ‘uniformly converge’?

Assuming that, we randomly sample $n$ data following a distribution, then if someone claims that the average of these $n$ data uniformly converge to its expectation with rate $O(\sqrt{1/n})$. Here, ...
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33 views

Find the MLE given the following probability

Data $x_1, . . . , x_n$ are modelled as observed values of i.i.d. random variables $X_1, . . . , X_n$ with each having common probability density function given by; ...
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79 views
+150

Techniques for proving asymptotic normality by Taylor expansion?

Suppose I have a sequence of densities $$ f_{X_n}(x) = \exp[\ell_n(x)], \qquad (x \in A). $$ My goal is to prove a statement like $T_n = \sqrt n (X_n - \mu) \to N(0, \sigma^2)$ in distribution, for ...
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8 views

Uniformly minimum variance unbiased estimator

How to prove $ \overline{X}=\frac{1}{n}\sum_{i=1}^nX_i$ is the uniformly minimum variance unbiased estimator of $\mu$ when $X_i\sim N(\mu,\sigma^2),$ and $\sigma$ is known. Idea: Let ...
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10 views

Interaction term (second order term) always implie the linear term (first order term)? (GLM)

I am doing a GLM model with interaction terms, but the same question can be asked with an ANOVA model. Suppose I have two independent variables $X_1,X_2$ and $Y$ the dependent one. I notice that, ...
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23 views

Proof of Central Limit theorem - infinite points

In the following proof I understand that we have taken power n because for sum of variables we take product of characterstic functions. Intuitively I understand why $n \to \infty$ is important but ...
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18 views

Prediction Intervals for Discrete Data

I would like to generate a prediction interval as described [here]. I suspect that my data comes from a normal distribution, but it was discretized into bins. How would one construct prediction ...
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5 views

How to parameterize some emprical data

I would like to describe a bunch of data that I have collected as a function of two variables. The data is phytoplankton absorption in my local area that has changed in concentration. The data looks ...
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1answer
13 views

Estimating population variance

Why is population variance estimated to be $\frac{1}{N-1}\Sigma_{1 \leq i\leq N}(x_i-m)^2$ as opposed to sample variance which is $\frac{1}{N}\Sigma_{1 \leq i\leq N}(x_i-m)^2$, where m is the mean? I ...
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1answer
19 views

Interpreting high p value and low correlation value

I am trying to run regression on financial data in R. I am new to regression analysis so I am finding it to difficult to interpret certain scenarios. I have the code as follows: Regression analysis ...
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1answer
13 views

Asymptotic test statistic for exponentially distributed data

I need an idea for tackling the following problem: Let $X_1,...,X_n\sim\mathrm{Exp}(\lambda)$ be an iid sample. We want to test $H_0:\lambda=\lambda_0$ vs. $H_1:\lambda\not=\lambda_0$ for some ...
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1answer
21 views

Transformation of a random variable.

In this transformation question, I managed to show the g(w) as given, yet I do not quite understand why the domain of the w is within plus and minus infinity. Shouldn't it be -1< w <1 as we ...
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1answer
49 views

Example of a real-world situation where multivariate analysis is applicable.

I have searched a lot of site to understand the situation where multivariate analysis is applicable. But not got any easily understandable example. Would you please give me a real-world example where ...
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20 views

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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1answer
33 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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1answer
31 views

calculate expectation of MLE

It's a question about whether $\hat{\theta _{MLE}}$ is an unbiased estimator of $\theta$. n independent pairs $(X_{1},Y_{1}), (X_{2},Y_{2}),....(X_{n},Y_{n}), n\geq 3$, where $Y_{i}=\theta ...
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1answer
26 views

how to create unbiased estimator in uniform distribution

$X_{1}, X_{2},...X_{n},n\geqslant 2, $ is a random sample from unif[$\theta -1, \theta +1$] Followed with the problem, I got T(X)=($X_{(1)}, X_{(n)} $) is sufficient but not complete, But I got ...
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9 views

Hypothesis testing, find $n$ to limit type II errors

I'm trying to solve the following (basic) problem, but I don't seem to get the last part 3). The average lifespan of a certain mechanical part is set to be at least 2 years by experts. The ...
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1answer
25 views

computing expectation of two arm bandit

assume you have a two arm bandit with one arm having a fixed, known probability of payoff $p = 0.6$ and another having an unknown payoff $q$, which is drawn uniformly from $[0,1]$. Each game the ...
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1answer
21 views

Calculate size and power of a given PMF

Let $X$ be a random variable having probability mass function $f(x) = \begin{cases} \dfrac{2+4a_1+a_2}{6}, & \text{if $x=1$} \\ \dfrac{2-2a_1+a_2}{6}, & \text{if $x=2$} \\ ...
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1answer
11 views

Expectation of an estimator

I want to find the expectation of $\hat\theta$. I have the cumulative distribution of $\hat\theta$: $$\Pr{(\hat\theta>t)} = e^{n(\theta-t)}\quad \text{for $t>\theta.$}$$ Now to find the ...
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2answers
33 views

Likelihood Theory and finding maximum likelihood

Let $X_1,\,\ldots,\,X_n$ be independent random variables, each with probability density function $$f(x;\,\theta)=\frac{2x}{\theta^2}\qquad \text{for }0<x<\theta.$$ I want to find the maximum ...
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1answer
46 views

Find joint likelihood function of observations $x_1, x_2, \ldots, x_n$ and $y_1, y_2, \ldots, y_m$

Let $x_1,\ldots,x_n$ be observations from a normal distribution with mean $0$ and s.d $s_1$. Similarly let $y_1,\ldots,y_m$ be observations from a normal distribution with mean $0$ and s.d ...
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1answer
7 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 ...
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43 views

$E\bigl(\frac{2}{1+x}\bigr)$ for Beta(2,$\frac{1}{2}$) random variable

Let x ~ Beta (2,$\frac{1}{2}$). Then calculate $E\left(\frac{2}{1+x}\right)$. So, ${E}[g(X)] = \displaystyle \int_{-\infty}^\infty g(x) f(x)\, \mathrm{d}x$ . $\displaystyle f(x;\alpha,\beta) ...
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1answer
18 views

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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11 views

Standard deviation of a sampling distribution

I have a question regarding the following video: https://www.khanacademy.org/math/probability/statistics-inferential/hypothesis-testing/v/hypothesis-testing-and-p-values From time 4:13 in the video. ...
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14 views

Aditive Outliers in Seasonal Time Series

I have adjusted a $SARIMA(1,0,1)(0,1,1)_{12}$ for my serie. Trying to improve my model I have detected with SAS aditive outliers in MAY and JUNE. My question is how to solve this outliers because if ...
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2answers
53 views

Optimal solution to a statistical decision problem

Setup I'm trying to find condition(s) that characterize the solution to a statistical decision problem. The environment is as follows. $\Omega$ is a finite set of states of the world. A decision ...
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1answer
44 views

Estimate the number of trials needed to observe all the possible outcomes of an experiment [duplicate]

I am stuck with the following problem: Each package of Pokemon cards contains 1 of N possible legendary Pokemon. How many packs do you expect to buy to get all N? We assume all N legendary cards are ...
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12 views

Doubt about the degree of freedom on standard error

Once I learn that the sample variance should be $s^{2}=\sum(X_{i}-\mu)^{2}/(N-1)$, I was told that "N-1" is came from the degree of freedom, even I didn't understand well the reason. Now, I am doubt ...
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How to represent data graphically in which a respondents can choose more than one group?

I am analyzing a survey in which one of the questions was (What is the industry segment of your company?) and the respondents can choose more than one category. I have used a pie chart to represent ...
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20 views

Difference among the same distribution , identical distribution and similar distribution.

$X\sim N(\mu_1,\sigma)$ and $Y\sim N(\mu_2,\sigma)$ are similar but not identical. $X\sim N(\mu,\sigma)$ and $Y\sim N(\mu,\sigma)$ are identical. But what is same distribution? Do same and ...
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110 views

Improvement of Minimum description length (MDL) estimate.

I earnestly request apology if this question is inappropriate for the forum. The question has two parts one technical and the other is not technical. I would appreciate any response. Let me consider ...
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1answer
82 views

Sampling via SRSWOR and biasedness of estimates

In a survey to estimate the proportion "p" of votes that a party will poll in an election, the voter list is divided into male and female lists. A sample of 100 from each list by simple random ...
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1answer
27 views

Suppose $40\%$ of the population possess a given characteristic … What is the probability $44\%$ or fewer possess the characteristic?

I have the following question: Suppose $40\%$ of the population possess a given characteristic. If a random sample of size $300$ is drawn from the population, then the probability that $44\%$ or ...
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1answer
16 views

Correlation Query

I can find $\mu_1, \mu_2, \operatorname{Var}(Y_1) \ \text{and } \operatorname{Var}(Y_2)$ but I am not sure how to get the co-variance of $Y_1$ and $Y_2$ in order to find the correlation of $Y_1$ ...
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1answer
21 views

Moment generating function of a uniform random variable.

My first attempt to this question was to find the first few moments about the mean and try to rearranging the those moments to obtain the general function as desired. However, when I tried to ...
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18 views

Finding variance with method of moments

In part (a), I know Var(Ni)=p(p-1) but how do I find the variance of the estimator with this result? And for part (b) of this question, I have to clue on how to tackle this question. I don't get ...