Questions tagged [normal-distribution]

The normal, or Gaussian, distribution has a density function that is a symmetrical bell-shaped curve. It is one of the most important distributions in statistics. Use the [normality] tag for asking about testing for normality.

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31 views

Maximum Likelihood Estimation of a dataset

I am coding a Maximum Likelihood Estimation of a given dataset (Data.csv). The goal is to estimate the mean and sigma. Note that the log of the dataset is well approximated by a normal distribution. ...
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QQ plot looks like gaussian, but kolmogorov smirnov test suggests to reject null hypothesis

I am testing my estimation of maximum likelihood parameters using the normal equation for gaussian generalised linear model. I am trying 3 covariates and 4 parameters (i.e. includingintercept) and in ...
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Does the sum of discrete uniforms coverge to a discrete Gaussian?

Is there some analogous of the Central limit theorem for discrete uniforms and discrete normal distributions? To be more specific, let's say we have identical and independent random random variables $...
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Understanding the stochastic part of rank-based inverse normal transformation

I'm checking out some methods for rank-based inverse normal transformation in Python and found this: https://github.com/edm1/rank-based-INT/blob/master/rank_based_inverse_normal_transformation.py ...
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Regression in linear equation vs. distribution form

In most introductory textbooks (less 'mathematical') simple linear regression model is formulated as equation. i.e. $$Y_i = \beta_0 + \beta_1 X_{1 i} + \epsilon_i, \quad i =1,n $$ $$ \epsilon_i \sim \...
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Setting Choosing alpha for generalized extreme Studentized deviate ESD

I'm working on a S-H-ESD implementation and I'm struggling to set the alpha for the ESD. The suggested alpha everywhere is 0.05. Is there a way to calculate an alpha based on the expected percent of ...
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Ideal observer test: which root is the correct one?

Given two pdfs of the signal when a target is present p(x|K)] and where there is only nois p(x|H)] I need to find an analytical solution for the ideal observer test: meaning that I need to find x. ...
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Defining when Distribution is Normal

Say we now a Random Variable X is normal $X \sim N(\mu, \sigma) $ Then we know that : $Z= (X-\mu)/\sigma $ ~$N(0,1)$ I answered in another question that we know Y is normal if it can be obtained ...
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Estimating likelihood of sample given median & median absolute deviation

I have a random variable $X$ that I expect is normally distributed, with some outliers. I can sample approximately 10 values from this variable, estimate its mean $\mu$ and std deviation $\sigma$. ...
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Why isn't a gaussian kernel subject to the curse of dimensionality?

This has been bugging me for a while now. I understand from this answer why gaussian kernels are effective. But I can't wrap my head around the intuition of why the infinite dimensional feature map 𝜙(...
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Are there two distributions whose product equals a gaussian?

Are there two distributions $X$ and $Y$ over $\mathbb{R}$ such that the distribution of the product $XY$ follows a Gaussian distribution?
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Integration of (Phi(x)-Phi(y))^2d(F(x, y)

How to integrate the following? Thanks
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Am I allowed to create a cumulative frequency distribution from non-normally distributed data?

I have data gathered from a circadian rhythm experiment and I like to present this data as a percentile cumulative frequency distribution (CFD%), because I think this is a clean way to present the ...
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How do I calculate $s$ from $\mathbb P(X+Y>u\mid X<s)=q$

Suppose $X,Y$ are (not necessarily independently) normally distributed, how can one calculate the maximal limit $s$ that $X$ may reach such that the probability of the sum $X+Y$ overshooting a given ...
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Problem of interpretation of Matlab's pca [closed]

I have generated 500 realizations of a Gaussian random process (with 501 observations) whose random variables are standard and the covariance kernel is also gaussian. When I perform the Karhunen-...
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What is the distribution of a bivariate normal component conditional on the max of the other component?

Let $n$ be a large integer, and consider two independent multivariate Gaussian $n$-vectors $x, z$ with $x\sim\mathcal{N}\left(0,I\right),$ and $z\sim\mathcal{N}\left(0,\sigma^2 I\right)$. Let $y=x+z$. ...
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Sufficient statistic for Gaussian $AR(1)$

Question Does the Gaussian $AR(1)$ model, with a fixed sample size $T$, have nontrivial sufficient statistics? The model is given by $$ y_t = \rho y_{t-1}, \, t = 1, \cdots, T, \; \epsilon_i \...
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Gaussian sufficient statistic calculation

Consider the Gaussian model $$ Y_i = \beta + \epsilon_i,\, i = 1, \cdots, n,\; \mbox{where}\; \epsilon_i \stackrel{i.i.d.}{\sim} \mathcal{N}(0, \sigma^2), $$ parametrized by $\beta$, with known $\...
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Combining regression estimates by summing

I want to know if one can combine regression estimates from panel regressions when the new dependent variable is a sum of the dependent variables from previously estimated regressions. To be ...
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Intuitive explanation of “density generators”?

I was reading through Meucci's Risk and Asset Allocation (2005), when I happened upon the concept of a "density generator", which I have not been able to find good explanations for anywhere online, ...
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Martingale Difference Sequence

I saw that in the link on page 3 it is said $Y_t = e_t\cdot e_{t-1}$ is martingale difference sequence and dependent where $e_t$ is i.i.d with $N(0,\sigma^2)$ Could you provide me with the proof of it?...
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Is it a valid to use one sample to estimate the population parameter and to decide whether another observation is in the population?

I have two groups of samples. Sample A is obtained from population A and sample B is a mixture of unknown origin i.e. some observations of sample B may come from population A and some may not. How can ...
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1answer
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Do both Bootstrap with and without replacement create a distribution?

I'm having a "noisy debate" with colleagues about whether sampling without replacement can still create a distribution. Methodology: A bootstrap (iterative process where I calculate Somers' D for new ...
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Measuring the uncertainty of points along a trajectory

I have a number of 2D trajectories, and I want to be able to estimate the variance of each step along the mean trajectory. I thought that this might be suited to Gaussian process modelling, but then I ...
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Martingale Difference Sequence CLT

Could you provide me with the proof of the following: $$n^{-1/2} \cdot\sum_t a_{t-j}e_t$$ converges to normal distribution as $n$ goes infinity by martingale difference sequence CLT where $a_t = \...
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Error distribution for Huber Regression

For linear regression there's an assumption that error terms come from normal distribution. so that $Y = aX + b + \epsilon$, where $\epsilon$ has normal distribution with mean zero and certain ...
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If $X$ follows standard normal distribution, find the correlation coefficient between $X$ and $\Phi(X)$

If $X$ follows standard normal distribution, find the correlation coefficient between $X$ and $\Phi(X)$, where $\Phi(X)$ is the cdf of $X$. My attempt is: First we have to calculate $Cov(X, \Phi(X))$...
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Integral and expected value for multivariate distribution

for the last couple of days I have been struggling with a problem and I was hoping to get some help here. I have a function that looks as follows: $$ f(x,y,z)=\begin{cases} 0 & \text{if} \ (x*...
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Statsmodel GenericLikelihoodModel - subclass Normal distribution

I'd like to use stats model GenericLikelihoodModel to fit a normal distribution. The doc gives an example with a probit distribution: https://www.statsmodels.org/dev/examples/notebooks/generated/...
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1answer
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How do I sample N items from a list ensuring the samples are as similar as possible to each other?

Is there anyway to sample 6 numbers without replacement this list of 12 numbers ...
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42 views

Conditional probability density function (PDF) of bivariate normal distribution

Let $X$ and $Y$ have bivariate normal PDF with correlation coefficient $\rho$, i.e.,: $f(x,y)=\frac{1}{2\pi\sigma_X\sigma_Y\sqrt{1-\rho^2}}\exp{(-\frac{1}{2(1-\rho^2)}[\frac{(x-\mu_X)^2}{\sigma_X^2}+\...
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1answer
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How can I generate 2 sets of variables from different distributions with a correlation between them in r? [duplicate]

I am working in R and would like to generate 40 numbers from $\mathrm{N}(0,1)$ and another 40 from $\mathrm{Uniform}(0,2)$ with a negative correlation (for example: $r = -0.45$) between them. The ...
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How can I check if my data-frame is normally distributed in R?

I have a data frame with 7 columns that holds numerical and integer values where some columns, even though numerical, are binary values (e.g. a dummy variable for sex; $0=\text{male}$, $1=\text{female}...
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1answer
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Are the skew-normal distribution and the skew-Cauchy distribution heavy-tailed?

I think the title is self-explanatory. I understand that the skewness and the tail behavior of some distribution are completely unrelated as any symmetric distribution will have a skewness of zero ...
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Why was the letter z chosen in z-scores?

Why was the letter z chosen in the name of a z-score = (data value minus mean) / standard deviation and not any other letter? This might be related to the same letter z being frequently used for the ...
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1answer
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constant matrix times Multivariate Gaussian distribution?

Suppose I have a multivariate Gaussian distribution x and a constant matrix A. I know how to calculate the mean and covariance of Ax but how can I prove that Ax will also be multivariate gaussian??
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What is the meaning of assuming a special prior on regularization method

I have heard/read that L1 regularization assumes Laplacian prior, however L2 regularization assumes Gaussian prior. But what exactly "assume" mean here? How does it work? How do each of these ...
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Conditional Maximum likelihood estimation and Normal distribution parameters

I understand that if I have a random variable $X$ and some data $D=\{x_1,x_2,x_3,\ldots\}$ (let’s assume that $X$ is normally distributed). I can use MLE to figure out the distribution parameters (...
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1answer
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Substracting two normal distributions [duplicate]

I was wondering if we have two normal distributions of X,Y~N(0,1), why is then X-2Y~N(0,5)? I understand the mean of the X-2Y distribution, but why is the variance 5?
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Posterior predictive distribution example

Assume there's some normally distributed population ($X$) whose parameters ($\mu$, $\sigma$) are not known. A sample ($x_1$) of size $n$ is drawn from $X$, and statistics are calculated: $\bar{x}_1$ ...
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Normal versus student-t distribution

I estimated two GARCH models, one with the Normal distribution and one with the Student-t distribution. The conditional volatility shows less noise for the model based on the Student-t distribution. ...
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Normality testing with very large sample size

Hypothesis testing such as Anderson-Darling or Shapiro-Wilk's test check normality of a distribution. However, if the sample size is very large, the test is extremely "accurate" but practically ...
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Hypothesis testing based on the concentrations of two cigarettes' samples

A cigarette producer sent two samples, one for each laboratory, which he thinks to be identical. The laboratories determined the concentration (in mg) of nicotine in each sample and they obtained the ...
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How to check if means and variances are normally distributed?

From here I read that The t-test assumes that the means of the different samples are normally distributed; it does not assume that the population is normally distributed. By the central limit ...
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How to normalize data by mapping data points from one mixture of multivariate normal distributions to another mixture

How to normalize data by mapping data points from one mixture of multivariate normal distributions to another mixture Problem description I am trying to normalize multivariate time series data. The ...
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1answer
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Value of normal distribution

Can anyone tell me what will be the exact value of zα at a 95% confidence limit?? zα is the standardized normal variable with (1 − α) confidence level. I am using this equation to find limits for Q-...
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MLE of joint normal covariance matrix under the constraint that the covariance is rank deficient

Given $n$ samples of a joint normal distribution of dimension $d$ with zero mean, $n>d$. I would like to ML estimate the covariance matrix, under the condition that the covariance is of rank $r<...
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When comparing timings from two algorithms, how sure can I be about which one is faster?

I'm doing some reinforcement learning (a type of machine learning), and I have measured how many episodes it takes my algorithms to reach a satisfactory performance level in a simulated environment. ...
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Comparing Gaussian and Poisson GLM when applied to count data; “Chi-Squared Error”

I have a fixed set of predictors ($[x_1,x_2,...,x_p]$), which I'm using to fit a GLM for univariate responses ($y_1$, $y_2$,...) of various types. E.g. I fit a GLM for $y_1 \sim [x_1,x_2,\dots,x_p]$, ...
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How to (and if) transform and standardize two types of data (count proportions & low-value inflated latencies) for a MRIM model

I am looking to analyse a variety of traits in a Multivariate random intercept model (MRIM) with the help of the MCMCglmm package in R. All traits are measured on different scales so I wish to ...