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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Jensen's inequality in derivation of EM algorithm

I am going through the derivation of EM algorithm and got stuck on understanding the following steps: Notes showing EM algortithm derivation For the equality to hold, f(x) has to be an affine ...
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22 views

Sampling bias using addresses as frame

A selected address in the sample has no residents. Will this cause bias in the results of a survey where house addresses are used as the frame? I think it wouldn't, because an address with no ...
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48 views

How to compute the $p$ value? and the correct explanation of the overall experiment.(Is my answer correct?)

Hello community first of all thanks for helping me with my math problems. Here I'm again with hypothesis test exercise. I want to know if I made some mistake in my answer and if someone can help me ...
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1answer
17 views

finding UMVUE $P ( X_1> t)$ [on hold]

Suppose $ X_1,\ldots,X_n$ are i.i.d. random variables with density: $$f(x_i;\theta)=\theta x_i^{-2}$$ $$x_i>\theta$$$$\theta>0$$ The smallest order statistic $X_{(1)}$ is sufficient and ...
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20 views

How to derive mean and variance for a Bayes estimator?

Let $X_1,...,X_n \sim$ iid $\mathcal{N}\left(\theta , \sigma ^2\right)$, where the variance is known. Also, suppose the prior distribution $\theta \sim ...
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18 views

Proving Weak Law of large numbers by Markov's inequality

Hi I am trying to solve the problem 5.13 of the book Statistical inference by George Casella and Roger L. Berger. The problem is Formulate and prove a version of the WLLN with a weaker assumption ...
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11 views

Estimate ratio of two expectations by sample means

I have a question about the estimation of a ratio of two expectations. Suppose $X_{i}$ and $Y_{i}$ are two random variables with $i=1,\cdots,N$. We seek to estimate $\mathbb{E}X_{i}/\mathbb{E}Y_{i}$ ...
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20 views

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

I am posting a similar question - in the previous one I put a wrong distribution, which changed the whole question. Let $X=(X_1,...,X_n)$ be i.i.d. $U(0,\theta)$. How to show that ...
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30 views

IMPROVED - Proving that a statistics is not sufficient (Gaussian case).

Let $X=(X_1,...,X_n)$ be i.i.d. $N(0,\sigma^2)$. 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 ...
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27 views

Use Maximum Likelihood Estimation to guess which dice got selected

We have two six-sided dice (faces numbered 1 through 6) and two tetrahedral dice (faces numbered 1 through 4). Someone selects two of them and throws each once. Then they tell us the sum of the ...
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Show that $\hat{\mu}$ has minimal variance

So two independent analyses of a content in a water sample have been made using two different methods, both without systematical errors but with different standard deviations. Method $B$ is assumed to ...
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Cramer-Blackwell estimator for uniform distribution.

I've got two estimators of parameter $\alpha$ in the distribution $X=X_1,...,X_n$, where $X_i$s are i.i.d. uniform random variables on the interval of $(0,\alpha)$. These two estimators are: ...
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28 views

Likelihood function for a distribution with both discrete and continuous components

Suppose $X_1, X_2, \ldots, X_n$ are $IID$ normal RVs with mean $\mu$ and variance $1$. However, we observe only $Y_i$'s where $Y_i = \max (0, X_i)$. I would like to know how to write likelihood ...
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23 views

Is Bayesian Inference what I need? [closed]

I'm not sure if I need a mathematician or a developer (or both) for this question. 1)There is a framework called Infer.NET that uses Bayesian inference for probabilistic programming. 2) I'm ...
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25 views

If $X$, $Y$ are random variables such that $E(X\mid Y=y)$ is constant for all $y$, then show that $E(XY)=E(X)E(Y)$ [i.e.,$\text{Cov}(X,Y)=0$] [closed]

If $X, Y$ are random variables such that $E(X\mid Y=y)$ is constant for all $y,$ then show that $$E(XY)=E(X)E(Y)\qquad \text{[i.e. Cov}(X,Y)=0]$$
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Likelihood Function for the Uniform Density. $ (\theta-1,\theta+1)$

Let the random variables $X_1,X_2,...,X_n$ iid $U[\theta-1\,,\theta+1]$. So the likelihood function therefore has the form: $L(\theta|X)=\prod_{i=1}^nf(X_i|\theta)=\frac{1}{2^n}I(X_1, . . . , X_n ...
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25 views

Calculate the estimators of $E[X]$ and $Var[X]$ using the method of moments

$(X_1,\dots, X_n)$ is a random sample extracted from a uniform distribution on the interval $$(\alpha-\beta, \alpha+\beta) \ \ \ \ \alpha \in \mathbb{R}, \beta \in \mathbb{R}^{+}$$ Demonstrate ...
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47 views

The mathematics of Correlation is not equal to Causation

In statistics, it is a common practice to say that "correlation does not mean causation", and mostly the proof for this is given by examples. While that is good for the intuition, it's not rigorous. ...
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21 views

Moment Estimator

I have been given Independent sample variables $X_1,...X_n$ that have a common p.d.f $$f(x,\theta)=\frac{10\theta^2}{x^2},$$ where $0<\theta<x$. How do I go about in finding the moment ...
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25 views

Showing that $S_{xx} = \sum_{1}^{n} x_i^2 - \frac{(\sum_{1}^{n} x_i)^2}{n}$

I am having problems understanding the identity (more specifically the last equality) $$S_{xx} = \sum_{1}^{n}(x_i - \bar{x})^2 = \sum_{1}^{n} x_i^2 - \frac{(\sum_{1}^{n} x_i)^2}{n}. $$ I've made an ...
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38 views

Maximum likelihood estimator of $\lambda$ and verifying if the estimator is unbiased

$(X_1,...X_n)$ is a random sample extracted from an exponential law of parameter $\lambda$ Calculate the likelihood estimator $\nu$ of $\lambda$. Then, if $n=2$: establish if $\nu$ is a unbiased ...
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19 views

What does this symbol mean for the MLE method?

This is an implementation of the maximum likelihood method on $\hat \pi$. I am unsure what that $\mathbf 1$ looking symbol means. MLE estimate of $\pi_y$ is ...
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10 views

statistics of membership, a simple case

Suppose we have two lists of real numbers $S_1$, $S_2$ and a particular real number $x$, which we are not sure which group it belongs to. Are there any tests in statistics that predict the membership ...
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20 views

Essential of statistical inference Solution?

i'm really interested in statistical inference, and I would try to learn it with this book : http://www.amazon.com/Statistical-Models-Cambridge-Probabilistic-Mathematics/dp/0521734495 However I can't ...
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25 views

Solving For Joint Likelihood In This Case

Let $Y_1,Y_2,\dotsc,Y_n \sim \operatorname{Poisson}(\lambda)$. Consider $U = \frac{1}{n} \sum_{i=1}^n Y_i$. Given the conjugate Gamma prior distribution $(\alpha,\beta)$, I want to show that the ...
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25 views

Asymptotic distribution of a measure of homogeneity

For an exam preparation I'm trying to solve the following question, but I get stuck. The question is One measure of the homogeneity of a multinomial population with $k$ cells and probabilities ...
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24 views

IID variables in statistics and real-life assumptions

IID (Independent and Identically Distributed) Random variables are often used in statistics, where a truly random sample is assumed to be made of IID variables. I'm studying basics of statistics (as a ...
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11 views

Normalized multivariate random vectors

While it is well-known that normalizing any zero-mean elliptically symmetric multivariate vector, such as a multivariate Gaussian, results in angular Gaussian distribution, what happens if the ...
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Comparing two textbooks for machine learning

I am a Ph.D student in Electrical Engineering. I am going to study the field of machine learning and I found some textbooks to study this field. 1) Probabilistic Graphical Models: Principles and ...
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Confidence Intervals

Let $x_1, \ldots, x_n$ be a sample from a normal population having unknown mean and variance. Let $\bar{x}$ be the average of the first n of them. What is the distribution of $x_{n+1} - \bar{x}$? If ...
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28 views

Hypothesis testing: Problem in finding the power of the test

Let X be a random sample of size one from $U(\theta,\theta+1)$ distribution, $\theta\in \mathbb{R}$. For testing $H_0:\theta=1$ against $H_1:\theta=2$, the critical region $ \left\{x : x>1 ...
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9 views

How to define a likelihood function for an EM algorithm

Assuming $A$ a set of vectors from a normal distribution, and $X$ a projection matrix and $B$ a set of projected vectors of $A$ using $X$: $B=A*X$ Using an EM approach and by initializing X from ...
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Probability that sample mean of iid Bernoulli random variables is close to true mean

Context: We have $X_i=1$ with probability $p$, $X_i=0$ with probability $1-p$, but $p$ is unknown. Given that the sample mean $Y_N=\frac{1}{N}\sum_{k=1}^N X_k$ is equal to $q$ after N observations, ...
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44 views

According to Chebyshev's rule, how many observations should lie within one and a half standard deviations of the mean?

Using the formula : $p = 1 - k^{-2}$ I calculated that $p = 1 - 1.5^{-2} = 0.56$ , which equals to $56\%$. Because I have $24$ data points I go ahead and solve the number of points is $56\%$ of ...
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Counterexample in Convergence in Distribution

I'm in my statistical inference course, and I've reached a problem related to convergence in distribution that I am slightly stuff on. Consider random variables $X_n,$ $Y_n$ who converge in ...
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Test for equality of two multivariate normal distributions for given µ and sigma

As title states, I'm interested in finding a proper inferential test to check if two multivariate normal distributions with given mean and standard deviation vectors have been draw from a common ...
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14 views

Variational inference on a Normal distribution: is my choice of priors passable?

I am trying to understand the basics of Variational Inference. In order to do so I designed a very simple problem: using the free-form mean field method to approximate the posteriori distribution of ...
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34 views

Reversing Central Limit Theorem?

I have a question like this. A company manufactures light bulbs. The life time of bulbs is assumed to be normally distributed. The CEO claims that an average light bulb lasts $300$ days. A ...
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19 views

Normal Distribution sample mean and population mean?

Assume that house prices in an area are normally distributed with a standard deviation of \$ 20,000. A random sample of 16 houses is taken. What is the probability that the sample mean differs from ...
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24 views

Approximation in Central Limit Theorem

I have a question like this. There are $50$ people in a line. The time takes to serve a person has a mean of $5$ mins and standard deviation of $3$ mins. $5$ people can be served at a time. What is ...
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33 views

Let $(X_1 ,Y_1),(X_2 ,Y_2),…,(X_n ,Y_n)$ be a sample from the uniform distribution on a disc $X^2 + Y^2 \leq \theta$, where $\theta$ is unknown.

Let $(X_1 ,Y_1),(X_2 ,Y_2),\ldots,(X_n ,Y_n)$ be a sample from the uniform distribution on a disc $X^2 + Y^2 \leq \theta$, where $\theta$ is unknown. That is, the joint density function of $(X,Y)$ is ...
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How do i know what's the sufficient statistic/estimator?

Let $X_1,X_2,\dots,X_n$ be a random sample from a distribution with p.d.f.: $$f(x, \theta)= e^{-(x-\theta)}; \theta<x<\infty;-\infty<\theta<\infty$$ obtain the sufficient statistic for ...
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15 views

Independent stochastic processes

I have 2 stochastic processes that are independent.. so E [X(t)C(t)]=E[X(t)]* E[C(t)] ... now I would know if ** X^2(t) and C^2(t)** are both independent and why.. Thanks
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Definition and use of Empirical Cumulative Distribution Function (ECDF)

Let $X_1 , X_2, \ldots ,X_n$ be independent identically distributed random variables with a common cdf $F(t)$. Then the empirical cdf is defined as , $$F_n(t) = \frac { \text{number of elements in ...
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63 views

For $n \geq 2$, let $X_1,X_2,\ldots,X_n$ be independent samples from $P_{\theta}$, the uniform distribution $U(\theta,\theta +1),\theta \in \mathbb R$

For $n \geq 2$, let $X_1,X_2,\ldots,X_n$ be independent samples from $P_{\theta}$, the uniform distribution $U(\theta,\theta +1),\theta \in \mathbb R$. Let $X_{(1)},X_{(2)},\ldots,X_{(n)}$ be order ...
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30 views

$X$ has a Poisson distribution and $Y$ has a Bernoulli distribution. Find a one-dimensional sufficient statistic for $\lambda$ based on $(X,Y)$.

Suppose we observe the pair $(X,Y)$ where $X$ has a Poisson$(\lambda)$ distribution and $Y$ has a Bernoulli$(\frac{\lambda}{1+\lambda})$ distribution, with $X$ and $Y$ independent and $\lambda \in ...
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Show that $P(X=-1)=p$ and $P(X=k)=q^2 p^k$, $k=0,1,2,…$ defines a probability distribution for the random variable $X$.

Let $p \in (0,1)$ and $q=1-p$. (a) Show that $P(X=-1)=p$ and $P(X=k)=q^2 p^k$, $k=0,1,2,...$ defines a probability distribution for the random variable $X$. (proved) (b) Given the single observation ...
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7 views

Interpreting data from a Gaussian Mixture Model using Gibbs sampling

I have data from a population with suspected subtypes within it. I have used a Gibbs sampler with different numbers of potential subtypes to produce Markov chains and posterior distributions. I am ...
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39 views

Find $a_n$ and $b_n$ such that $a_n (\max_{1 \leq i\leq n}X_{i} - b_n)$ converges in distribution to a non-degenerate random variable.

Let $X_1,X_2,...X_n$ be iid with the same chi-square distribution with one degree of freedom. Find $a_n$ and $b_n$ such that $a_n (\max_{1 \leq i\leq n}X_{i} - b_n)$ converges in distribution to a ...
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50 views

Example of using Delta Method

Let $\hat p$ be the proportion of successes in $n$ independent Bernoulli trials each having probability $p$ of success. (a) Compute the expectation of $\hat p (1-\hat p)$. (b) Compute the approximate ...