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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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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1answer
25 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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22 views

Is Bayesian Inference what I need? [on hold]

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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24 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.,$\text{Cov}(X,Y)=0$] [on hold]

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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2answers
23 views

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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1answer
24 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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1answer
46 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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1answer
24 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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22 views

how large a sample must be drawn-to make the probability $.95$ that a $90$ percent confidence interval for $\mu$ will have length less than $\sigma/5$ [closed]

In sampling from a normal population with both $\mu$ and $\sigma$ unknown, how large a sample must be drawn-to make the probability $.95$ that a $90$ percent confidence interval for $\mu$ will have ...
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1answer
37 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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31 views

The patterns of markov chain [closed]

I have one question let $X$ be a Markov chain that could take the values $1,2$ or $3$ with the same probability $1/3$. what is the probability that $(1,2,1)$ pattern occurs sooner than $(2,1,3)$?
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1answer
18 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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1answer
24 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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1answer
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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2answers
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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8 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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17 views

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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2answers
25 views

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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1answer
27 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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1answer
27 views

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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43 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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1answer
18 views

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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25 views

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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1answer
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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1answer
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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1answer
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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1answer
31 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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1answer
14 views

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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1answer
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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33 views

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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3answers
62 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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29 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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45 views

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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38 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 ...
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1answer
21 views

Determine the asymptotic distribution of $\bar X_n$, properly centered and $\sqrt n$ scaled

Let $X_1, X_2,...X_n$ be i.i.d. with $P(X_i =1)=1-P(X_i =0)=p,p \in (0,1)$. (a) Show that $\bar X_n$ is the MLE of p. (b) Find the mean $\mu_n$ and variance $\sigma^{2}_n$ of $\bar X_n$ and invoke the ...
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70 views

Theoretical distribution of a random variable

Martin has $n$ words, and he wants to make a computer program that chooses for him $k$ words (and shows them to him), where $k \le n$, for as many times as he clicks a button until all of the words ...
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54 views

Does a Markov process have memory zero?

I have the following question: The Markov process with two states and a transition matrix $$P =\begin{pmatrix} 0.3 & 0.7 \\ 0.3 & 0.7 \end{pmatrix}$$ has memory zero. Is it true? My ...
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48 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=\\ ...
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39 views

Bayesian information criterion from measure theoretic point of view?

Bayesian information criterion (BIC) is well known and it is derived from the maximizing the posterior density function which is equivalent to solving the marginal likelihood integral. My question is: ...
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1answer
29 views

How to test a collection of samples are sampled with replacement or not?

A box is full of balls with $m$ different colors, and for each color, there are $n$ balls. So the total number of balls is $m*n$. Note that $m$ is unknown, $n$ is already known, and balls can only be ...
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6 views

Adjoint of evaluation operator: Inverse Bayesian Analysis

I'm reading "Inverse Problems - A Bayesian Perspective" by Andrew Stuart and I'm stuck with working out an application (an easier form of section 3.2): Consider a random process $u: (0,1) \to \mathbb ...
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1answer
29 views

Do i get the right MLE and 90% confidence interval of normal distribution?

I think i do right in step1 above. But i wonder whether i get the right confidence interval of mu and sigma in step2?
2
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1answer
22 views

Two-sample permutation tests for earthquake magnitudes before and after a damaging quake

Background and data. An earthquake of magnitude 5.17 stuck near Yountville, California in the early morning hours of September 9, 2000, injuring about 25 people and doing about $50 million damage. ...