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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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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55 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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16 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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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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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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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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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
34 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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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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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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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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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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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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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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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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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 ...
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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 ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 = ...
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26 views

How do i find the sample standard deviation?

Question: The following data were drawn from a normal population:4, 8, 12, 11, 14, 6, 12, 8, 9, 5. estimate the population mean with 90% confidence. i understand how to go about the problem and i ...
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Shows Weibull distribution belongs to a one dimensional exponential family

It is given that $f_\eta(y) = h(y)exp(\eta T(y)-A^*(\eta))$ $P_Y(y)= \frac{k}{\lambda} (\frac{y}{\lambda})^{k-1}exp(-(\frac{y}{\lambda})^k)$ What i did was by arranging $P_Y(y)$ to get $\frac{k}{\...
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what's the z parameter in LCA formula?

In this paper on LCA, http://members.home.nl/jeroenvermunt/hagenaars2002b.pdf I can understand the basic eqn. in terms of disease (theta) with symptoms (y). $$ f(\mathbf{y}_i|\theta)= \sum_{k=1}^K ...
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Is there a way to know when a minority of the data is telling the truth?

I am working with temperature data obtained from a network, and i need to identify when temperature sensings are legitimate(Haven't been modified by someone). To find this out, i must compare new ...
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Conditions for using a t-test for means: contradictory or not?

Besides the conditions of independence and 10%, my teacher says that we need an approximately normal distribution before performing a t-test. My question is: Is this contradictory because the math we ...
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How many people should I ask if a statement (A) is true if the same can be inferred by asking two other statements (X and Y implies A)?

I am asking a number of participants if they believe a given statement is valid. I have a number of such statements, some of which can be inferred. In the made up example below, ...
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Necessary to find an estimator's probability distribution before calculating its expectation?

Where $X_{1}, X_{2}, \dots X_{n}$ is an iid distribution with pdf given by: \begin{cases} \frac{1}{\theta}x^{1-\theta} \qquad &\text{If $0 \leq x \leq 1$} \\[5 pt] 0 \qquad &Otherwise \end{...
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1answer
28 views

Hypothesis testing: normal vs. non-normal

I have the following hypothesis testing problem: $$H_0:X=Y,\quad\text{vs.}\quad H_1:X=Y+Z$$ where $Y\sim\mathcal{N}(0,\sigma^2)$ and $Z$ is a random variable with non-normal continuous ...
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23 views

Conceptual/Notational question on conditional distributions and “given”

So in the book I'm reading, I see the notations $f(x|\theta)$ being used to refer to population distributions, dependent on $\theta$ which are in a family. The author explains this as a notational ...
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The Cramer-Rao Lower Bound proof

Let $X_1, . . . , X_n$ be i.i.d. with density function $f (x|θ)$. Let $T = t (X_1, . . . , X_n)$ be an unbiased estimate of $θ$. Then, under smoothness assumptions on $f (x|θ)$, $$Var(T) >= \frac{1}...