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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27 views

Distribution of the sum of the two dependent bivariate gaussian distributions and related questions

This is something I was thinking about and I decided to modify a question from a mid-term to ask this. Suppose $X_{1}$ and $X_{2}$ are two bivariate gaussian variables, decribed as $$ X_{i}=\begin{...
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6 views

The derivative of the absolute value |x| [migrated]

I read about the derivative of the absolute value |x|, but why the absolute value is not differentiable at point zero, and when it becomes 1 or -1 {geometrically}? Thanks
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16 views

Joint distribution of density forecasts

I have a panel data set and I have created a model and finally I have obtained some density forecasts. That is, I run my model for the $y_{it}$ and i obtain predictions for $\hat{y_{i,t+1}}$ , $\hat{...
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10 views

What is the conceptual difference between the marginal variance and the range in the Matérn covariance?

The Matérn covariance kernel is given by: $$ C_\nu(d) = \sigma^2 \frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}\frac{d}{p}\right)^\nu K_\nu\left(\sqrt{2\nu}\frac{d}{p}\right) $$ My question is, what ...
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22 views

transform on a distribution

I have a pseudo-normal distribution with mean 0 and sd 0.03. Is there a way to transform this distribution such as values above or under +/- one standard deviation are pushed towards 1/-1 while values ...
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0answers
17 views

2d points distribution with shifted centroid

How can I have a distribution with centroid being away from center? i.e. points distribution starts out dense, and then goes on to become sparse, but with NO sudden transition? Pl find the image ...
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2answers
39 views

Compute standard deviation of a normal distribution given a mean and a range or width?

This is sort of an odd question, I realize - but it has to do with random number generation. What I'd like to do is generate random numbers with a normal distribution; many functions (in Python) do ...
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19 views

Methods and Demographics

Specific question: How do statisticians determine the likelihood that such-and-such type of person will commit a crime? What is their methodology? To illustrate what I'm talking about, people will ...
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16 views

Compute a Quadratic discriminant analysis (QDA) in R assuming not normal data and missing information

In this course, the professor is saying that we can compute a QDA with missing data points and non-normal data (even if this assumption can be violated). But the problem is that I don't know any ...
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1answer
29 views

How to make sense of non-linear data transformations? What conclusions drawn can you apply to original data?

In stats class, the professor talked about the interest of transforming skewed data sets to make them more "normal". From what I've understood so far, the idea is that the normal curve has nice ...
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15 views

Pointwise multiplication of two Gaussians and normalization in Python

The Bayesian update contains a multiplication of the Prior and the Likelihood. The area under the bell curve of a gaussian sums up to 1. We know that this only holds if we integrate from -Inf to +Inf. ...
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12 views

Splitting Gaussian into N beta distributions [on hold]

Suppose we have a standard,symmetrical Gaussian distribution. The extreme values are not very likely, the central values most likely. Is there a method to split this gaussian distribution into $N$ ...
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20 views

Python Bayesian invgamma.rvs - joint posterior of normal distribution sampling

(My question is inspired by this blog post: The Bayesian analysis of normal distributions with Python. If you read it, you will get a good background on what I am asking.) I am trying to model the ...
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1answer
25 views

estimating variance using only data at the tails without resorting to Gibbs sampling

Suppose we know that the population size is $n=1,000$ but for whatever reason, we only have the bottom $n_1=100$ observations and the top $n_2 = 200$ observations. Furthermore, suppose we know the ...
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1answer
62 views

Variance computed using Taylor series does not agree with numerical experiment

I would like to estimate an angle $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ given the noisy observations of its sine and cosine (this is related to my earlier question). My estimator is ...
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1answer
29 views

What is the best way to approximate probability from a PDF of a Gaussian distribution? [closed]

In a program I am writing, I have a Gaussian Distribution function that returns the PDF given a specific vector. The issue is, this is obviously not the actual probability. To further complicate ...
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43 views

How can I check the quality of my probabilities?

I have a data set where the number of accidents are given together with the probability (assumed to be correct) that the number of accidents is less than 4. The task is to calculate the probabilites ...
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33 views

Fit a Gaussian to data with R with optim and nls

I want to fit a Gaussian to the following data: ...
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37 views

Gaussian Process in single dimension

Suppose that I start with a two-dimensional (zero-mean) Gaussian process; following Rasmussen and Williams, I denote it by $$ f(x, y) \sim \mathcal{GP}(0, k(x, y, x', y')), $$ where $(x, y) \in \...
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1answer
59 views

Derived Distribution from normal distribution

\begin{align} X_{1} \sim N(\mu_{1} , \, \sigma_{1}^2 ) \\ X_{2} \sim N(\mu_{2} , \, \sigma_{2}^2 ) \end{align} Assume $X_{1}$ and $X_{2}$ are independent, what is the distribution of $ Y = 1/X_{1} ...
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36 views

How to integrate lognormal equation?

The problem comes from the personal research $h \sim \log N(\mu,\sigma^2)$ then $$f_H(h) = \frac{1}{h\sigma\sqrt{2\pi}}\exp\left[-\frac{1}{2}\left(\frac{\log h-\mu}{\sigma}\right)^2\right].$$ Here ...
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10 views

What error distribution can I use for GLMM with continuous data but not normal due to too many 0s?

I am having problems with building a generalised linear model with random effects. I am modelling how a sensitivity ratio between various taxa and cyanobacteria (logSR) is effected by the taxa and ...
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22 views

What error distribution can I use for GLMM

I am having problems with building a generalised linear model with random effects. I am modelling how a sensitivity ratio between various taxa and cyanobacteria (logSR) is effected by the taxa and ...
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2answers
46 views

Is it possible to estimate the standard deviation of a normal distribution if I only have the mean of the population?

I'm not a math or statistics expert and only have a self-taught basic understanding of these things. I'm working on a problem where I know the mean of the population and I want to estimate the ...
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10 views

Confusion regarding conditional distirbution of the product of gaussian process with a normal

I am getting confusedregarding a simplae problem. say $Y = WX$ where $X$ is a q X n matrix with each row a Gaussian process denoted as $\mathcal{GP}(M(\mathbf{X_i}),C(\mathbf{X_i},\mathbf{X_i}))$, in ...
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Steve Hsu's calculation of geniuses in China

On his blog, physicist Steve Hsu wrote the following: Assuming a normal distribution, there are only about 10,000 people in the US who perform at +4SD and a similar number in Europe, so this is ...
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11 views

How to smooth distribution with Gaussian kernel?

I have a theoretical cross section(just a function $f(p^2)$) which depends on $p^2$(where $p$ is impulse). I know that equipment give we $p_{exp}$ with normal distribution $N(p_{real}, \sigma)$. So, ...
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1answer
53 views

What will be the value of $a$ that maximizes $g(a)?$

Let $X$ be a normal random variable with mean $2$ and variance $4$, and $$g(a)=P(a\leq X\leq a+2)$$,Then what will be the value of $a$ that maximizes $g(a)?$ We can write,$$g(a)=\Phi(\frac{a}{2})-\...
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23 views

Fisher information matrix with a general covariance structure

For the linear model, general linear models which allow for a more general covariance structure $V(\theta)_{N\times N}=(I_{N}+\theta A_{N\times N})(I_{N}+\theta A_{N\times N})^{'}$ ,where $A_{N\times ...
3
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1answer
203 views

Exact multinomial goodness-of-fit test as a normality test

We have a practical real-life problem in an open source Linux related project. And I would like to hear an expert review/opinion about the way we are trying to solve this problem. It's been more than ...
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1answer
80 views

Please help me understand normal probability plots

I am a teacher that will be teaching about Normal Probability Plots. This topic is new to me and I would be grateful for any help. Essentially the specification for the course expects students to ...
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36 views

Generating samples from high-dimensional multivariate Gaussian with few training samples

Say I have a $n\times d$ dataset $D$ where $n\ll d$ ($n$ number of observations, $d$ number of dimensions). Currently, if I want $m$ samples from $D$ assuming it is multivariate Gaussian, I can do ...
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1answer
43 views

How would you calculate $E[\mid x \mid ^{\alpha }], \alpha \in \Re$?

Here $x \sim N(0,1)$. I realize that the expectation won't be defined for $\alpha$ when the integral goes to infinity. I can't seem to figure out which specific values of $\alpha$ would cause this. ...
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1answer
70 views

Distribution of Quotient of 2 dependent random variables

Well , I have the following problem.. Let $X_1,\cdots ,X_{2n}$ be iid $N(0,1)$ random variables. Define $$U_n=\left({X_1\over X_2}+{X_3\over X_4}+\cdots +{X_{2n-1}\over X_{2n}}\right)$$ $$V_n=...
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2answers
28 views

Mann Whitney U test with normal distribution approximation: null hypothesis rejected?

I'm new with U test and I have some doubts about the rejection of the null hypothesis with the U test with normal distribution approximation. In my example I used this data for a 1 tailed test: $$ ...
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2answers
51 views

How to show that $X_2$ is also a standard normal variable?

Suppose $X_1$ is a standard normal variable. Define, $$X_2=\begin{cases}-X_1, & \text{if } |X_1|<1,\\X_1, & \text{otherwise}\end{cases}$$ Show that $X_2$ is also a standard normal random ...
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133 views

Wavelet-domain gaussian processes: what is the covariance?

I've been reading Maraun et al, "Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation, and significant testing" (2007) which defines a class of non-stationary GPs that can be ...
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18 views

parametric or non-parametric in comparison of a dependent variable within several groups

I am comparing maths and chemistry scores of students in several schools. I am not comparing the schools with each other. I am just trying to say that students do better in maths than in chemistry in ...
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75 views

If I do a robust regression using standard error, what do I need to analyse in the residuals

Let's say I do a multiple regression, using robust (Stata option). It is a robust standard error regression. I want to analyse and discuss residuals. Residuals versus fitted values Is it ...
3
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2answers
95 views

Normal distribution necessary for linear-mixed effects? (R)

This is my first post on this site. I'm a linguistics graduate student who is struggling to grasp the basics of statistics. I've run a questionnaire in which participants had to rate sentences from 1 ...
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21 views

Robustness of Gaussian Copula

I am looking for studies regarding the robustness of a multivariate Gaussian copula. Specifically I am wondering whether estimates of the dependence parameter in a multivariate Gaussian Copula (Sigma) ...
3
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1answer
40 views

Conditional expectation of a univariate Gaussian

Suppose I have a univariate Gaussian distribution with mean $\mu_X$ and standard deviation $\sigma_X$, and I know the random variable $X$ is least some positive value $y$: $X \geq y$. What is the ...
3
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1answer
72 views

How to find $E|X+Y|^3$ from related information?

Assume that $$ E(X+Y)=E(X-Y)=0 $$ $$ V(X+Y)=3 $$ $$ V(X-Y)=1 $$ Show that $E|X+Y|\leq\sqrt3$. If in addition, it is given that $(X,Y)$ is bivariate normal, calculate $E|X+Y|^3$. For the 1st part, ...
3
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58 views

How do I perform Bayesian Updating for a function of multiple parameters, each with its own distribution?

I have a variable that is a recursive function involving other variables with known distributions (see problem below). Let $b(t+1) = b(t) + C \sqrt{b(t)}$ where I know $C \sim N(1.82, .0298)$ and ...
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2answers
68 views

Covariance matrix for a linear combination of correlated Gaussian random variables

Supposing $X$ and $Y$ are random variables with a joint bivariate normal distribution and covariance matrix $\Sigma_{XY}$. Consider the following linear combination for constants $A$, $B$ and $C$: $$...
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1answer
26 views

Does Naive Bayes assume normality?

I came across this paper about Naive Bayes that states [Naive Bayes] is based on another common simplifying assumption: the values of numeric attributes are normally distributed within each class. ...
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25 views

Normal distribution analog for HSV colour space

[This question inspired by work by Jason Thornton et al (see https://cryptome.org/2012/05/person-large-area-spy.pdf , Equation (4))] I am interested in modelling a distribution over the HSV color ...
2
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1answer
91 views

Impossible to bring to normality

I have given a data and I have to check the data if it's normally distributed and if not I have to transform the data into normality. I had done shapiro-wilk normality test and p-value is clearly ...
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1answer
19 views

Does all variables in a VAR/VEC need to be normally distributed, or only the target variable?

Well? Does all variables in a VAR/VEC need to be normally distributed, or only the target variable? It is very hard to get all of them to meet criteria of normality without deleting too many outliers.
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1answer
22 views

Probability distribution arising from the combination of a normal variable and an other random variable

Let $X\backsim N(0,5^2)$ and Y be an independent random variable taking the values +1 and -1 with equal probability.Find the distribution of $S=XY+\frac{X}{Y}, T=XY-\frac{X}{Y}$ I have solved the ...