Tagged Questions

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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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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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Weighted variance of a small sample

I am trying to calculate the variance of a small sample. I have the data: ...
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Confidence interval for difference in means

I need to obtain a 95% confidence interval for the indifference in the mean score overall I have the following data which states subject (A-L) and then Test and Retest A B C D E F G H I J K ...
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Is the statement: $p\left(\left.y\right|h^{-1}\left(\varphi\right)\right)=p\left(\left.y\right|\varphi\right)$ correct?

Say I have a likelihood function $p\left(\left.y\right|\theta\right)$ and I make the reparameterization $\varphi=h\left(\theta\right)$ using the bijective function $h$ with inverse $h^{-1}$. Then it ...
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Doubts on Bayes hypothesis test

I meet one problem on hypothesis testing in statistic theory. "Assume given the probability spaces $(X,S,\mu_i)$, $i=1,2$. $H_i$, $i=1,2$, is the hypothesis that $T$ is from the statistical ...
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Doubts in Bayes' Theorem

I meet one problem on the probability and statistic theory. "Assume given the probability spaces $(X,S,\mu_i)$, $i=1,2$, and the probability space $(X,S,\lambda)$. And there exsit functions ...
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Monte carlo formula to compute the approximation of variance of MLE

In the book of "Monte Carlo Statistical Methods", the book gives an approximation formula for the variance of MLE, Later on, the book mentions that this approximation formula can be written as ...
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Nonrejection region- equivalently the area determined by confidence interval for $H_0: \hat \beta=\beta^*$

For $H_0: \hat \beta=\beta^*$ I want to prove that the non-rejection region in level of significance approach Will be eqaul to the area determined by upper and lower bounds in confidence interval ...
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Regarding jointly multivariate normal X1,X2…X5

So I have a question from statistical inference that I need some help with: $X_1,X_2,...,X_5$ are jointly multivariate normal with means = $\mu_i$, variances = $\sigma^2_i$, correlation = $\rho$ ...
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Determining constant in a CDF

I have a question that I literally have no idea how to begin, I was hoping someone could help me: It says $X_1,X_2,\ldots,X_n$ is a sample from a distribution It says that the Cumulative ...
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Gaussian Approximation of an intractable distribution

I am currently encountering this problem: I have an intractable distribution and I want to minimize the KL divergence of this distribution and a multivariate gaussian distribution. So we just need ...
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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 ...
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What is the most general formalism for machine learning?

Most of the literature I can find in the field of machine learning is extremely practical, listing many techniques you can use like neural networks, SVMs, random forests, and so on. There are lots of ...
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Finding MLE of $f(x;\theta) =1$ if $\theta-1/2<x< \theta+1/2$

Let $X_1,...,X_n$ have density: $f(x;\theta) = \begin{cases} 1 &\mbox{if } \theta-1/2<x< \theta+1/2 \\ 0 & otherwise \end{cases}$ Let $Y_1=min \lbrace X_1,...,X_n \rbrace$ and ...
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Find MLE of $\alpha$ of $f(x;\alpha)=(1+\alpha x) /2$ (stuck at derivative setup)

$X_1,...,X_n$ is an independent sample with common density: $f(x;\alpha)=(1+\alpha x) /2$ where $-1<x<1$ and $-1<\alpha <1$ I have to find the maximum likelihood estimate of $\alpha$. ...
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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 ...
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Calculating that confidence that pairs of lightbulbs are independently illuminated.

So, you're sitting in a dark room, and on the far wall you see $n$ lightbulbs mounted above plaques numbered $1$ through $n$. There is a lightswitch on the arm of your chair. Every time you flip the ...
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Interval overlap maximisation problem

Consider a line of equally spaced sensors and a disturbance which travels unidirectionally along the line at a fixed speed such that the disturbance takes time $\tau$ to travel between adjacent ...
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Estimating the total attendance

Suppose you do not know how many people are attending a convention, but you do know that as each person entered he was given an identification tag with a number on it. The tags are numbered serially ...
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Jensen inequality

Does Jensen inequality, which is $\mathbb{E}(g(x)) \geq g(\mathbb{E}X)$ if $g$ is convex, assume that $\mathbb{E}X$ (expected value of random variable $X$) must belong to $R(X)$ (range of random ...
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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 ...
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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 $\quad x_1,x_2,\ldots,x_n$ ...
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Central Limit Theorem (multivariate)

It's known that a sum $\mathbf S_n$ of iid random vectors $\mathbf {X_1,X_2, X_3,...X_n}$ which are $\mathbb{R^d}$ normaly distributed with covariance matrix $\Sigma$ and mean $\mathbf 0$, will ...
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Fun with Powerball history. What probability theories apply?

I wanted to play around with probability theories and see if I find any statistical inferences from the Powerball lottery history going back to 11/1/1997. What theories could I apply besides simple ...
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A representation theorem for a minimally sufficient statistic by Bahadur

The Statement of the Problem I'd appreciate help in proving the following, unproven theorem from a classic article by Bahadur ([BAH], Theorem 6.3) (the expressions in square brackets are my ...
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What is the probability that we get more than $\frac{n}{2} + 2\sqrt{nln(n)}$ heads? [duplicate]

Toss $n$ coins. What is the probability that we get more than $\frac{n}{2} + 2\sqrt{n[\ln(n)]}$ heads? How do I apply Chernoff Bounds to this? I really need help understanding Chernoff Bounds.
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Two definitions of Bayes Sufficiency

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

I have some survival times which are exponentially distributed for two groups G1 (treatment) and G2 (control). The data are censored with a censoring distribution given by h(c), so I only observe: 1) ...
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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 ...
How can I show the following necessary and sufficient condition? An unbiased estimator $\hat{\theta}$ of $\theta$ achieves the Cramér-Rao Lower Bound if and only if $$\frac{\partial ... 2answers 311 views Taylor series approximation statistics how can I show the following: Let X_1, X_2,\ldots, X_n be i.i.d Poisson with mean \lambda. Let Y = |\{i: X_i =0\}|. Then \lambda is estimated by$$\eta = - \log(Y/n)$$Use Taylor series to ... 1answer 40 views independence of uniform random variables1 let X_j \sim U(0,1) if$$Y_j=\frac{X_j}{X_1+X_2+\cdots+X_n} I want to show that: $Y_j$are independent $\operatorname{Var}(Y_1)=\dfrac{c}{n^2} +o\left(\dfrac{1}{n^2}\right)$ then calculate ...
I have a question about joint convergence results derived from an FCLT (i.e., a Functional Central Limit Theorem). To motivate my question, consider the following setup: Let $y_t$ be a random walk ...