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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30 views

Making sense out of the method for finding posterior distributions.

I have been recently studying Bayesian statistics and more precisely the problem of finding posterior distributions. I am able to understand the my textbook's problems, but I realize that I understand ...
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1answer
57 views

Probability that a clumsy boy eats $k$ out of 20 candies

A week or two (or maybe more) ago, the following question was posted and then deleted just as I was getting to the end of my solution. Unfortunately I have now forgotten what my solution was going to ...
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17 views

How to best compare two different time series with different frequencies

Lets say I have two time series $X_t$ and $Y_{t,q}$. As an examples, lets say $X_t$ is a series that measures year over year changes in the level of output of a good (say number of widgets). So $X_t = ...
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1answer
427 views

How do I calculate a point estimate of the largest 10%?

Here is the provided data: The question asked is: Calculate a point estimate of the value that separates the largest 10% of all values in the thickness distribution from the remaining 90% and state ...
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1answer
2k views

Rolling standard deviations

I am trying to calculate standard deviations on an array of numbers. My psuedo code looks like this: ...
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2answers
45 views

Sum over Binomial mass function

In Casella and Berger Book (Statistical Inference), exercise 2.40 is $$\sum_{k=0}^x {n\choose k}p^k(1-p)^{n-k}=(n-x){n\choose x}\int_0^{1-p}t^{n-x-1}(1-t)^xdt.$$ If I replace $x$ by $n$ then LHS ...
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12 views

find out min/max of statistical distribution (GPA) from median, mode, count, size of elements? [closed]

I would like to find to the min/max of a distribution given the following. Was wondering if it is possible. You could think of them as GPA Number of elements in distribution: 93 Theoretical min of ...
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11 views

Invariance to measurements, and invariance to a group of transformations

Why would one demand invariance to measurements(IM) when, previously they've already assumed invariance to a group of transformations(IG). Isn't the IM a special case of IG? Any help would be ...
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3answers
38 views

Expected number of sides of a dice

I have two dice, one with m sides (labeled 1,2,...m) and one with n sides (labeled 1,2,...n). I roll both three times. The m-sided one comes up 1, 2, 9 and the n-sided one comes up 7, 7, 8. Which is ...
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0answers
11 views

What ranking system should I use for my website? (Wilson vs. ELO/Glicko)

So I have users rate stuff on my site, and so I want to put the "highest-rated" stuff at the top and "lowest-rated" at the bottom. As now, am I only using positive and negative ratings, but this could ...
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2answers
846 views

connection between PCA and linear regression

Is there a formal link between linear regression and PCA? The goal of PCA is to decompose a matrix into a linear combination of variables that contain most of the information in the matrix. Suppose ...
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7 views

Precision-Recall Graph: F1 Score v.s. Break-Even Point

To evaluate two classifiers from the aspects of Precision-Recall, two measures are often used: F1 score and Break Even Point (BEP for short. I failed to find any document about it from wiki, and it is ...
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14 views

Inferring the addends of the sum of two random variables

I have three independent Poisson variables: B, C and D, whose parameters $\lambda_B$, $\lambda_C$ and $\lambda_D$ are unknown. I sample once the variable: $$ A_1 \equiv 0.9\cdot B + 0.1\cdot C $$ and ...
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1answer
35 views

How to combine correlated estimates to test variable is > 0?

Let X1 and X2 be two unbiased but correlated Gaussian estimators of a true value x. 1. What is the proper way to combine two observations of X1 and X2 to test whether x > 0? 2. How does the answer ...
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1answer
72 views

On characterization of MRE estimators

I have some trouble understanding the second equality in the proof of theorem 6;
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2answers
351 views

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 ...
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0answers
17 views

Improving Probability of Event='1' in Logistic Regression (SAS)

I will try to give as much background as I can and if more is needed I will gladly give more. I'm working on trying to find an equation that will tell what probability you have a defaulting on a loan ...
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1answer
34 views

Finding the confidence interval of a normally distributed sample

Traffic police monitor the speed of vehicles as they travel over a new bridge. The average speed for a sample of 27 vehicles was 91.29 km/h, with the sample standard deviation being 4.94 km/h. We will ...
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11 views

Stuck on a step in the derivation of the variance of a sample variance

I am trying to understand the formula for the variance of a sample variance $$var(S^2_n) = \frac{1}{n} \left [ \mu_4 - \frac{n-3}{n-1}\cdot \sigma^4 \right ] $$ We start from: $$ var(S^2_n) = E(S^...
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1answer
693 views

Kruskal Wallis - Effect size

I analyse 4 algorithms and 3 sets of metrics for each algorithm in which I apply the non-parametric Kruskal-Wallis test for each metric to detect any differences in performance between these ...
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0answers
14 views

Bayesian equation: need for priors

As far as I understand, in the problem of Bayesian inference we have a random variable $y$ describing data, which is distributed according to some parameter $x$ via the known conditional distribution $...
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1answer
26 views

How to inference the conditional probability about LDA?

I'm studying the paper of Blei, "Latent Dirichlet Allocation" ( http://www.jmlr.org/papers/volume3/blei03a/blei03a.pdf ). In his paper(page 1003), given equation is $p(\theta, z|w, \alpha, \beta)= \...
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17 views

Cramer roa lower bound for complex numbers

I calculated the Cramer-Rao bounds on variance of these parameter: $VAR_\gamma>\frac{(1-|\gamma(\omega)|^2)^2}{2N}=\sigma_{|\gamma(\omega)|}=\frac{1-|\gamma(\omega)|^2}{\sqrt {2N}}$ I would like ...
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0answers
14 views

Find the critical value in Tukey's HSD

I'm trying to find the formula for finding the critical values for Tukey's HSD but I can't find any documentation on how to calculate the critical value based on the number of groups the type I error ...
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0answers
11 views

Integrate Beta and Normal CDF mixture

Is it possible to integrate the following integral? $\int_0^1 y^{m-1}(1-y)^{n-1}\Phi\left(\Phi^{-1}(y)+\mu\right)dy$, where $m, n, \mu$ are constants and $\Phi(.)$ is the normal CDF Thank you
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1answer
73 views

On randomized estimators [closed]

I been reading the following text on randomized estimators, I cant manage to understand how the randomisation is incoparated into the randomized estimator. How does the random mechanism fit in, ...
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9 views

Reference Books on Asymptotic theory of Statistics and Probability

Can anyone suggest me some good reference books on Asymptotic Theory of Statistics and Probability for students pursuing a post-graduate degree in Statistics ? It would be very much helpful if the ...
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1answer
27 views

Using the Central Limit Theorem to calculate a mean from Poisson distributed random variables

Firstly, I am studying the basic concepts of statistics and so any explanations, advice and suggestions are more than appreciated. Onto the problem- I am given the central limit theorem and understand ...
2
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1answer
24 views

questions about 2 sample t-tests

So I'm just a bit confused about 2 sample t-tests and just want to write out what I think I know and see if that's correct, so if anyone could tell me whether or not what I'm writting is true that ...
2
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1answer
123 views

Confidence level of random sample from continuous distribution

Let $X_1,X_2,\cdots ,X_n$ be a random sample from a continuous distribution with median $\mu$. If $[X_{min}, X_{max}]$ is used as a confidence interval for $\mu$, what is its confidence level? What is ...
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17 views

Is this an exponential family of distributions? from casella and berger 6.20

I am trying to do 6.20 in Casella and Berger part d. The solutions manual says that the order statistics are minimal sufficient and not complete. I understand their logic, but why doesn't this work? ...
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73 views

Finding the right $\sigma$-algebra. Question on uncertainty related to the secretary problem.

I'm working on a problem related to the secretary problem. Let me give a short overview on the topic I research: You are supposed to choose the best item presented to you in a row of n items. Any ...
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27 views

Guessing Mathematical Probabilities by Tests

I'm stuck with a (maybe simple) problem. I have 4 values possible for a test, and I can do as many tests as I want. What is the minimum number of tests required to be at least at 95% sure I have the ...
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1answer
22 views

Relation between estimator's consistency and biasedness

I have two quick question: If an estimator is consistent, does that imply it is unbiased? If an estimator is biased, does that imply it is not consistent? we know that consistency means ...
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1answer
7 views

Missing approximation to get the Maximum A Posteriori (MAP) estimator of event times with a sparse prior

Assume that a signal $ y $ is a noisy perturbation of time-shifted copies of a given waveform $ f(t) $ defined on K time bins $ \{ 0, \cdots, K-1 \} $: \begin{equation} \forall t \in \{1, \cdots, T\}...
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1answer
13 views

Maximum a Posteriori (MAP) Estimator of Time Shifts with Poisson Process Prior

Assume that a signal $ y $ is a noisy superposition of time-shifted copies of a given waveform $ f(t) $ on a finite time interval $ [0, T] $: \begin{equation} y(t) = \sum_{i=1}^{n} f(t - \tau_j) + \...
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1answer
62 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 $24$:...
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19 views

How should I calculate the MLE based on a random sample from $PAR(\theta,2)$

Consider a random sample of size $n$ from a Pareto distribution, $X_i \sim PAR(\theta, \kappa =2)$. I have to compute the MLE, $\hat \theta$, to three decimale places. So I started doing the ...
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77 views

Expectation or Integration of the normal cdf

Can any one help me how to solve this pronbelm? I have a random variable $W$, i.e., $$W=\Phi(X)^k\Phi(-X)^m=P(Z\le X)^kP(Z \ge X)^m,$$ $X$ is Normal($\mu$,1), $Z \text{ is Normal(0,1)}$, and $k$ ...
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0answers
32 views

Finding independence of two variables

I am trying the following problem: Let $(X_1, Y_1)\ and\ (X_2, Y_2)$ be random points on the plane such that $X_1, X_2, Y_1, and\ Y_2$ are independent $N(µ, σ^2)$. Let $D^2\ $ denote the squared ...
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1answer
24 views

sequential anova r

I am a really confused. Assume we have a multiple regression model: $$ y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} +...+ \beta_{k}x_{k} $$ Using R we can make a test: $$ H0: \beta_{1} = \...
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18 views

Finding Asymptotic Confidence Interval with a condition

I am trying to solve the following problem: Let $X_1, X_2$, and $X_3$ be random variables from the following joint pmf: $$f_{X_1,X_2,X_3}(x_1, x_2, x_3) = \frac{n!}{x_1!x_2!x_3!} p_1^{x_1} ...
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1answer
951 views

Improper Uniform Prior Distribution

In Bayesian method, choosing the prior distribution is an important step when using the Bayesian method. When choosing prior, we consider the prior knowledge to choose which prior distribution is the ...
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48 views

How to calculate the probability that $X_n$ is not the largest observation in the sample?

I am trying to solve the following problem: Let $X_1,\dots, X_n$, where $n > 4$, be independent random variables such that $X_i ∼ N(i, i)$ for $i = 1, \dots, n$. Let $\bar{X} = {\frac{1}{n}}{\...
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18 views

finding the sufficient and ancillary statistics

I am trying to find the sufficient and Ancillary statistics for the following problem: suppose $(X_1, Y_1). . . ,(X_n, Y_n)$ be iid random vectors from the pdf: $f_{X,Y} (x, y) = {\frac{1}{2π \...
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1answer
64 views

Product of two uniform random variables/ expectation of the products

Suppose I want the expectation, $E\Phi(X-\mu)\Phi(\mu-X)$, where $\Phi(.)$ represents the Normal CDF, and X is $Normal(\beta,1)$. Consequently $\Phi(.)$'s are uniform[0,1] and at the same time two ...
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1answer
76 views

Estimating the radius of a circle

I have a circle iwth radius $r$. I want to test the hypothesis that $r \leq 2$ vs. $r >2$ based on the posterior of $r$. $r$ follows the prior distribution: $f(r) = \frac{2}{r^{2}}$, $ r >0.5$. ...
2
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0answers
85 views

How to solve an integratation involved an unknown function?

Can anyone have any suggestions how to solve this equation for $w_i$, that is, what is the solution of $w_i$? $$ \int_0^\infty e^{\Phi^{-1}(w_i)ε_i}P(r_i│ε_i )f(ε_i )dε_i=δ $$ Where, $f(ε_i)$ is the ...
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28 views

How to find expectation of Binomial Mass Function?

For example, $$ E \scriptstyle\binom{n}{r}\Phi(X)^r(1-\Phi(X))^{n-r} $$ Where X follows normal distribution with mean $\mu $ and standard deviation 1, and $\Phi(.)$ is the normal CDF. Thank you
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1answer
22 views

Risk function for vectors

How do apply risk functions to vectors? Here is the problem I have encountered: Let $X = (X_1, X_2, . . . , X_p)$ be a collection of independent random variables with $X_i \sim N(\mu_i, 1)$ for $i = ...