The normal, or Gaussian, distribution has a density function that is a symmetrical bell-shaped curve. It is often used as a reference against which other distributions are compared.

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Infinite fourth moment and maximum entropy

Alright, I expect this is a silly question, but I don't actually know, so. Suppose there is some random variable that's distributed on the reals, and all I know about the distribution is its mean ...
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256 views

Which is kernel similar gaussian kernel?

I must find a kernel that statisfies as follows: In the my reference paper, the author suggest gaussian kernel that is The purpose of that kernel is that it will take a weight for each points ...
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79 views

$E(\frac{1}{1+x^2})$ under a Gaussian

This question is leading on from the following question. http://math.stackexchange.com/questions/360275/e1-1x2-under-a-normal-distribution Basically what is the $E\left(\frac{1}{1+x^2}\right)$ under ...
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19 views

Analyzing regression results

I have done a regression model where i determine the number of cubes (independent variable) based on the amount of units i started with for each product type (dependent variables, ...
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82 views

Proving some properties of expected first order statistics with respect to sample size

Question: Consider $n$ random variables $x_1, x_2,\cdots x_n\sim \mathcal{N}(0,1)$. The expected value of the $i$th order statistic (the maximum) can be written as $E(\mathcal{O}^n_1)= ...
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Normal Bayesian Model: Marginal distribution of $\bar X$ with unknown mean and unknown variance

For $i=1, \ldots, K$ and $j=1, \ldots,n$, assume the following model. \begin{align} X_{ij} \mid \mu_i, \sigma^2 & \stackrel{_\text{ind}}{\sim} N(\mu_i, \sigma_i^2) \\ \mu_i & ...
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19 views

When doing ANOVA, what do we need to assume is normally distributed? [duplicate]

Do we assume that the population distributions are normal, or that the sample distributions are normal, or that the sampling distribution is normal? If the latter, what do we mean by sampling ...
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1answer
72 views

Is it Poisson distributed, and if so, what's its meaning?

I've collected data from my website. The website is about cars. The data are about user reviews and the cars. what we see in the graphs is the probability of some car type (Ford Focus 2008) to have X ...
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2answers
25 views

Evaluate mixture model

I have a question concerning the evaluation of mixture models. Is there a gold standard to compute the goodness of a fit for a mixture model? What I am concerned about is how one would evaluate if ...
6
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1answer
82 views

The standard normal distribution vs the t-distribution

Given an IID normally distributed sample $X_1,...,X_n$ for $n$ small with mean $\mu$, standard deviation $\sigma$, sample mean $\overline{X}$ and sample standard deviation $s$ (the unbiased estimator ...
6
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118 views

What does standard deviation tell us in non-normal distribution

In a normal distribution, the 68-95-99.7 rule imparts standard deviation a lot of meaning. But what would standard deviation mean in a non-normal distribution (multimodal or skewed)? Would all data ...
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0answers
16 views

Sampling Distribution of mean for Poisson Distributions

Suppose I have random variables $X_i$ which are Poisson distributed with mean $\mu$. I m interested in the sampling distribution of the variable $\frac{X_1+...+X_n}{n}$. We now that as $n$ goes to ...
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1answer
108 views

Question with MLE

I'm having some problems with this question, and was hoping someone here could help. Let $X_1,\ldots,X_2$ be $n$ determinations of a physical constant $\theta$. Consider the model $X_i = ...
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1answer
104 views

What does it mean “being normally distributed”

There is an exercise which is used to illustrate how normal distribution works. The exercise starts by saying "Suppose scores on an IQ test are normally distributed..."; What does it mean for the ...
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1answer
18 views

Regression Gaussian Estimates instead of points

I am using a support vector regression is order to get estimates of a variable y. I want to receive a probability distribution of my estimates and not just point estimates. I want to predict Gaussian ...
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0answers
41 views

Quantifying Potential for Violence/Espionage

I am planning a seven-question Likert-type survey with a "1 to 10" scale. The questions I am asking in the survey are to assess an individual's risk factors (how much risk they present to an ...
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0answers
20 views

Combining independent Gaussian probabilities

I am using three Gaussian distributions with which I generate random numbers to represent many candidate xyz points. I use some selection criteria (details not particularly relevant) to decide on ...
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0answers
47 views

If $X, Y$ are jointly standard normal with correlation $r$, and $a, b$ are constants, what is $p(Y < b | X < a)$?

The application here is interpreting the correlation coefficient $r$ in terms of $X$'s ability to predict $Y$ for extreme values of $X$. For example, if $r = .8$, then what is $p(Y < 0 | X < ...
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28 views

How to get only positive values when imputing data?

Suppose age is normally distributed with mean 20 and standard deviation 5. How do you ensure that you get only positive values when you sample age from this distribution? I am trying to impute ...
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18 views

Marginalization in gaussian

If $\theta|\mu \sim N(\mu,\sigma_o^2)$ and $\mu \sim N(0, \sigma_1^2)$ what is the marginalized $P(\theta)$. $\theta$ and $\mu$ both are nx1 vectors $P(\theta) = \int P(\theta|\mu)P(\mu)d\mu$ Is it ...
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51 views

Method to identify samples lying outside the normal distribution

In my previous questions I was looking for a method identify samples that had a variability significantly higher than the rest of my dataset. Methods to determine reliability of measurements using ...
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1answer
47 views

Anomaly detection using exponential weighted moving average

I would like to detect anomaly using exponential weighted moving average. I don't have series of data points. All I have is EMA(t-1) and the data point of the current time(t) DP(t). From these data, ...
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1answer
41 views

Using F-tests for variance in non-normal populations

I'm fairly new to stats, so please excuse me if this problem is hopelessly elementary or misinformed. Basically, I'm wondering if you can help me understand whether I'm using the F-Test for variance ...
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77 views

Using extreme value theory to estimate bounds

Suppose I have I have a random variable $X$ that I know is doubly bounded on support $[0,\theta]$ but I dont know $\theta$ (we don't know anything on the distribution of $X$, but assume it is not ...
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28 views

UMVUE for normal distribution $\sigma$

Let $X_1,X_2,...,X_n$ be a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. I showed that $(\bar X,S^2)$ is jointly sufficient for estimating ($\mu$,$\sigma^2$) where ...
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75 views

Approximation of logarithm of standard normal CDF for x<0

Does anyone know of an approximation for the logarithm of the standard normal CDF for x<0? I need to implement an algorithm that very quickly calculates it. The straightforward way, of course, is ...
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139 views

Extreme Value Theory - Show: Normal to Gumbel

The Maximum of $X_1,\dots,X_n. \sim$ i.i.d. Standardnormals converges to the Standard Gumbel Distribution according to Extreme Value Theory. How can we show that? We have $$P(\max X_i \leq x) = ...
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83 views

Sum of normal truncated random variables

Suppose I have $n$ independent normal random variables $$X_1 \sim \mathrm{N}(\mu_1, \sigma_1^2)\\X_2 \sim \mathrm{N}(\mu_2, \sigma_2^2)\\\vdots\\X_n \sim \mathrm{N}(\mu_n, \sigma_n^2)$$ and ...
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48 views

Confidence interval for the standard deviation of a Normal distribution with known mean

Suppose $\textbf{Y} = (Y_{1}, ... , Y_{n})$ is a random sample from the $N(\mu, \sigma_{0}^{2})$ distribution where $\mathrm{E}(Y_{i}) = \mu$ is unknown but $\mathrm{SD}(Y_{i}) = \sigma_{0}$ is known. ...
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1answer
25 views

How to implement a 2-D Gaussian Processes Regression through GPML (MATLAB)?

I just touched Gaussian processes two weeks ago. I am not very familiar with the selection of a model and its hyperparameters. Here is the demo code that I run for a 2-D Gaussian processes regression. ...
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20 views

How to calculate $\int\limits_{-\infty}^{+\infty} \frac{n-1}{\alpha}\Phi(\frac{x}{\alpha})^{n-2}\phi(\frac{x}{\alpha})^2dx$? [migrated]

I was working on a research project that involves taking the integral of $\int\limits_{-\infty}^{+\infty} \frac{n-1}{\alpha}\Phi(\frac{x}{\alpha})^{n-2}\phi(\frac{x}{\alpha})^2dx$, where ...
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1answer
70 views

Difference between standard deviation and standard deviate

I was reading this paper http://comjnl.oxfordjournals.org/content/20/4/359.full.pdf and in the last paragraph of the first page, Rule 1 is given as $\alpha_{j+1} > \bar{\alpha} + k s_{\alpha}$. ...
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228 views

Is the beta distribution really better than the normal distribution for testing the difference of two proportions?

I'm working at an online agency, where we run a lot of AB testing in order to test differences in proportion between two groups (test vs. control). Standard practice in the industry for testing ...
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27 views

Does normal distribution assumption improve the quality of results of a meta-analysis of effect-sizes?

Different methods of meta-analysis have different assumptions and one of the popular assumption that has been invoked so often is that of normal distribution. can we work out meta-analytic results ...
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93 views

Is there a name for this process/ distribution?

Does the equation below have a name, or is it similar to some other well-known process/ equation? Equation of interest: $$S_c = S_{c-1} + S_{c-1}\omega_c\delta_c$$ $\delta\sim\mathcal{N}(0,1)$ is a ...
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1answer
24 views

Scale parameter MLE scheme known but how to find according distribution PDF?

For known location, we can find the scale parameter of a normal distribution by calculating the sum of squared differences to the location, then dividing by n-1 and taking the square root. This is the ...
2
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1answer
50 views

Absolute variable as dependent variable

I have the following model: $$ |X| = B_0 +B_1 \cdot y + B_2 \cdot z , $$ where $z$ and $y$ are normally distributed random variables, and $B_1$ and $B_2$ denote the coefficients. My dependent ...
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1answer
16 views

Does a time-series have to be stationary before you calculate a z score or t score?

It's been a long time since basic statistics. I have a financial time-series that exhibits exponential growth. Before I standardize, do I have to make the time-series stationary? Before I ...
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1answer
71 views

Maximum likelihood estimation for mixed Poisson and Gaussian data

Background I've been doing a little bit of work lately on maximum likelihood estimation (MLE), for cases where the data is normally-distributed and also for cases where the data is Poisson ...
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26 views

Centering vs. Standarizing which one is better? [duplicate]

Two approaches have been proposed in order to overcome the issue of multicollinearity if we have interaction variables which are mean centering and standardizing (z scores). You can check No.2 in this ...
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1answer
48 views

Determining Whether Values Within One Standard Deviation Meet Some Condition

Suppose I have a set of objects, each of which can take on values within some unknown range, and each of which only has one sample of 50+ values associated with it. The values of each observation is ...
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1answer
45 views

bayesian logistic regression - gaussian distribution on parameters?

I'm trying to read this article about Bayesian logistic regression. In general, to classify instances, they use: $p(y=+1 |\beta) = \sigma(\beta^TX) $ (where $\sigma$ is obviously the sigmoid ...
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14 views

Is Monte Carlo simulation more appropriate than parametric tests for constructing confidence intervals for weighted means?

A colleague suggested that Monte Carlo simulation should be preferred for constructing confidence intervals for weighted means calculated from a sample. How and why exactly might Monte Carlo perform ...
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1answer
58 views

Understanding the Chi-squared test and the Chi-squared distribution

I am trying to understand the logic behind chi-squared test. The Chi-squared test is $\chi ^2 = \sum \frac{(obs-exp)^2}{exp}$. $\chi ^2$ is then compared to a Chi-squared distribution to find out a ...
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1answer
22 views

Frequent Occurrence of Rotated Graph of Logistic Function

I have found that in much of the data that I am looking through, after sorting from largest to smallest, there is a pattern similar to a rotated logistic function. That is, it declines steeply along ...
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Is it ok to transform a logarithm variable to z score

I have a variable that has 57 kurtosis, so I decided to transform it to log. However, I have multicolleanirity problem due to interacting this variable and others with another variable so I am using z ...
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14 views

Normal Data Distribution (Visual tests accept and statistical test ks reject normal distribution) Need Help [duplicate]

I have four variables want to run regression while i check for data distribution i found that histogram and qq plot provide evidance of normal data distribution where as ks test is significant for all ...
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26 views

GEV of Normal Distribution and relationship of the parameters

My question goes on Extreme Value Theory for the Normal distribution (www.math.ethz.ch/~embrecht/RM/chap7.pdf): Which type of GEV (Generalized Extreme Value) distribution does the Normal ...
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2answers
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Match Right Skewed Distribution to Normal

I am running a simulation. One of my parameters is sampled from a normal distribution. I would like to perform a sensitivity analysis using a right skewed distribution. This is what I had hoped to ...
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60 views

Bayesian model with unknown mean and variance with lognormal prior

For $i=1, \ldots, K$ and $j=1, \ldots,n$, assume the following model. \begin{align} X_{ij} \mid \mu_i, \sigma^2 & \stackrel{_\text{iid}}{\sim} N(\mu_i, \sigma^2) \nonumber \\ \mu_i & ...