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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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Summation of two Gaussian distributed data with different coefficient of mean and variance

I need some help on Gaussian distribution. i have two dataset, both are identical and independent distributed, but having mean as 2μ_1 and μ_2, same scenario for the variance. How can I add them? ...
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2answers
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Can left-censored data be normal distributed?

Very often I read that the distribution of IQ-test results is normal. However, the IQ-scale is left-censored, i.e. test results cannot be lower than zero. The normal distribution, however, is usually ...
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probability involves bivariate gaussian

I'm working on a spatial project. I need to calculate the probability of a point being the closest to another. Say I'm given four points $y$, $x_1$,$x_2$ and $x_3$ in 2D plane, and let $Y'=y+Z$, where ...
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1answer
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Recovering a distribution after Gaussian noise is added

I have a large dataset (400k rows) in which I suspect the data has been obfuscated by the addition of a Gaussian distribution. My guess is that some of the data had categorical variables (based on the ...
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Normal Distributions Definition [on hold]

Confirm the bivariate Gaussian mean and covariance by explicitly evalu- ating the exponential integral using the definition of the multivariate Gaussian density function
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26 views

What is a good aproximation in asymptotic normality?

I have a conceptual doubt. For example, suppose I have $X_i \stackrel{iid}{\sim} N(\theta^*,1)$ and I know that (I have the information) $\theta^{*}\geq 0$. So I have the Constrained Maximum ...
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1answer
27 views

P-Value in Shapiro-Wilk test

I've read that if this p-value is less than 0.05 (for a 95% confidence interval), the null hypothesis that the data comes from a normal distribution must be rejected. However, in the following ...
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Updating the variance of a Normal Distribution Using Bayes [on hold]

I have a prior belief for the mean $\mu_p$ and std $\sigma_p$ of a normally distributed variable $X$. (no dataset). So I have the mean of these parameters, but NOT their variance (I'll just presume ...
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1answer
51 views

Distribution of Maximum Likelihood Estimator

Why is the Maximum Likelihood Estimator Normally distributed? I can't figure out why it is true for large n in general. My attempt (for single parameter) Let $L(\theta)$ be the maximum likelihood ...
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2answers
36 views

What is the assumption on the distribution of data in gaussian mixture models?

I am reading about Gaussian mixture models from this slide https://www.ics.uci.edu/~smyth/courses/cs274/notes/EMnotes.pdf However, I am super confused at the very first line. It says: We ...
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1answer
50 views

$E[\bar{X^3}]$ of N(μ,1)

Suppose $x_1, x_2, x_3,\ldots, x_n$ i.i.d. Normal(μ , 1) random variables with μ $\in$ $\mathbb{R}$ how can we calculate $E[\bar{X^3}]$
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Male-female height difference estimation from thresholds

Introduction. Assume that two populations $A$ and $B$ are distributed with normal distributions $N(\mu_A, \sigma_A^2)$ and $N(\mu_B, \sigma_B^2)$. This is a general problem, but as an example, I will ...
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1answer
33 views

Higher order moments of a multivariate Gaussian rv

Let $X~N_d(\mu,\Sigma)$ be a multivariate Gaussian random vector. Is there a convenient formula for each of $$ \mu_p\triangleq \mathbb{E}\left[\sum_{i=1}^d |X_i|^p\right], $$ in terms of $\mu$ and ...
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2answers
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Proving $\Gamma\left(\frac{1}{2}\right)=\sqrt\pi$ using the expected value of standard normal variable

I'm looking to prove that $\Gamma\left(\frac{1}{2}\right)=\sqrt\pi$ using the fact that $E(Z^2)=\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}}e^{\frac{-z^2}{2}} z^2\, dz$ (where $Z$ is a standard ...
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1answer
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Normalizing two independent weights in order to produce output between 0 and 1

I have two scores, alpha and beta, ranging both between 0 and 1. I want to weight these with weight_one, weight_two in order to favour one of these scores over the other. Then, afterwards, I want to ...
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Determining sample size of a Gaussian subset with new std

I have N1 samples that is distributed close enough to Gaussian with 0 mean and std1. For simplicity, we can assume N1=1000 and std1 = 7. Now I want to choose a subset of those samples that is ...
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Transforming GPS location datapoints in R

I'm very new to this kind of analysis so I hope my question is of good interest. I've GPS locations from three independent GPS loggers, which data consists of two variables: latitude and longitude. ...
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2answers
58 views

Derive multivariate from bivariate normal distribution

Could anyone help me on the following. Let $K$ and $M$ be integers so that $K\geq3$ and $2\leq M < K$. Let $\boldsymbol{X}=(X_1, ..., X_K)^T$ be a random vector, $\boldsymbol{\mu}$ be a $K\times 1$...
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1answer
93 views

Sampling from Gaussian mixture models, when are the sampled data independent?

Suppose I generate a Gaussian mixture model with $N$ Gaussian distributions $p(x) = \sum\limits_{i = 1}^N w_i \mathcal{N}(x;\mu_i, \Sigma_i)$ where $w_i$ are the weights. Now I sample some points $\...
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1answer
57 views

Does the peak of a Normal Distribution mean anything? [closed]

What does the peak of a Normal distribution show? Let's say if I have a flat peak, does this mean I have a larger variance? What if I have a sharp peak? For example, Does the "blue distribution" ...
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Error on distribution of values with uncertainties

I'm currently analysing a measurement which results in a high number of values, each having a slightly different uncertainty. These values are following a gaussian distribution, so I wrote a python ...
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0answers
37 views

Poisson vs Gaussian GLM: which to use?

I'm going to provide a simulated case. However, the question is of a general nature (see end of the post). Let's suppose we have some data generated in this way: ...
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2answers
171 views

Double integral involving the normal CDF

I need to compute (or best approximate?) the following integral $$\int_0^\infty \int_0^\infty (1 + \alpha u)^{-1}(1 + v)^{-1} \Phi\left(\frac{\beta}{\sqrt{\gamma + uv}}\right) \text{d}u \text{d}v,\...
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2answers
31 views

Fitting a partial distribution to a normal distribution

So, I've got a(n assumed) normal distribution that is left bound by 0, and I'm trying to extrapolate what would happen if the distribution could go negative. Given a portion of the right tail of a ...
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0answers
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How to calculate the Product between Gaussian and Gaussian distribution in Matlab? [duplicate]

I require to calculate the Product of Gaussian distributed and Gaussian distribution. My work clamp force = (force of the wheel * friction coefficient) where the force of the wheel is Gaussian ...
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2answers
521 views

Why use a Gaussian mixture model?

I am learning about Gaussian mixture models (GMM) but I am confused as to why anyone should ever use this algorithm. How is this algorithm better than other standard clustering algorithm such as $K$-...
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0answers
30 views

Confusion with Computing Probabilities of a Normal Distribution without the Integral

How does this code is calculating the probability of Normal distribution without calculating the integral ...
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2answers
404 views

Inverse of the covariance matrix of a multivariate normal distribution

Is the covariance matrix of a multivariate normal distribution always invertible?
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15 views

BIC score for Gaussian Mixture Model

I'm not sure how to compute a BIC score for multiple classes. For exmaple , I have a supervised problem with 3 classes.I fit 3 gaussian using MLE. Then, if I want to compute BIC score: I have to ...
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1answer
40 views

Deriving standard normal distribution from a statistic involving normal and uniform random variables

I tried deriving distributions of numerator and denominator separately. But found that there is no closed form. I have no clue on how to show that Z is standard normal.
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1answer
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Sample size for the normal approximation of the Binomial distribution

I am reading about the familiar hypothesis test for proportions, using the normal approximation for large sample sizes. Some people say (write) that the condition for using the approximation is np>5 ...
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1answer
27 views

Which statistical test to use on half-normal distribution of discrete data points

I have two datasets. Each contains integer data points, ranging from 0 to 34 and they follow an approximately half-normal distribution, as 0 occurs very frequently and 34 occurs only once. I would ...
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1answer
40 views

Select optimal points for Gaussian process with a well-known target function

I'm currently trying to select the optimal points for a Gaussian Process Regression, and the important thing is that i already know the whole target function. Therefore, it's not Online Learning ...
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Are these confidence intervals calculated correctly?

Is the 1SD calculation correct in this image? Instead of 16% for 1SD on each side of the mean, they have 34%. SD2 and SD3 are correct. SD2=100-95 = 5%. Divide by 2 = 2.5%. and same for SD3 = 99.7% (....
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1answer
12 views

Calculating normality for a two way ANOVA with repeated measures

I am trying to calculate if my residuals are normally distributed for my two way ANOVA with repeated measures on Rstudio. However, all the tutorials I find seem to specific to one way within ANOVAs ...
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21 views

Is it possible to experimentally reproduce the probability of type 1 errors in significance tests?

In some fields, a lot of sample studies feature statistical tests. Oftentimes, box plots of sample distributions are shown. I was wondering whether i could read distributional parameters from these ...
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How can I calculate $\int{f(\bar{x}|\theta)g(\theta)d\theta}$ when $\bar{X}$ and $g$ are both normally distributed? [duplicate]

I'm reading Berger & Sellke (1987) 1. On page 115, the following statements are given: $$ \begin{align} m_g(x) & = \int{f(x|\theta)g(\theta)d\theta} \\ g & \sim \mathcal{N}(\theta_0,\...
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1answer
30 views

Computing posterior density of Normal with unknown $\mu$ and $\sigma^2$

(From Bayesian Essentials with R by Marin & Robert page 31) We are given an iid sample $\mathfrak{D}_n = (x_1, \dots, x_n)$ from the normal distribution $\mathcal{N}(0, \sigma^2)$ and $\theta=(\...
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0answers
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How to calculate $\int_{S}^{T} \Phi \Big(\frac{l_0-f(t)}{\sigma}\Big)g(t)dt$? [migrated]

How can I calculate the integral- $\int_{S}^{T} \Phi \Big(\frac{l_0-f(t)}{\sigma}\Big)g(t)dt$ where $f(t)= \mu_0+\mu_1 e^{-\gamma (7+logt)^\delta}$ and $g(t)= \frac{1}{(t+h)^k}$ . Here $\Phi \Big(\...
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1answer
39 views

How do you normalise a histogram with two peaks?

I have a histogram that looks as such and I want to use it as part of a Linear Discriminant Analysis but the lda requires its variables to have a normal distribution. What kind of transformation ...
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5answers
2k views

Why Normality assumption in linear regression

My question is very simple: why we choose normal as the distribution that error term follows in the assumption of linear regression? Why we don't choose others like uniform, t or whatever?
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1answer
40 views

Does standard distance follow the 68-95-99.7 rule?

I'm wanting to do a simple standard distance demonstration for my students in R, but I've come across a conundrum. When I simulate the creation of 10,000 points in a spatial normal distribution, ...
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0answers
25 views

How can one variable show difference between samples, when the previous didn't? ( according to t-test)

I don't have Statistics background, and I'm trying to learn and perform some tests. I work in oligonucleotide purification, and I have two machines for that purpose. They should release the exact same ...
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1answer
37 views

Twenty samples should be the same. But How should I test that

I have a table derived from a group of polypeptides or proteins. Because they are proteins they are made up of 20 amino acids. Iff a group of proteins contains a random sample of amino acids then the ...
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1answer
52 views

Variance of Normal Order Statistics

Suppose we have $X_1, \cdots, X_n \overset{\textrm{i.i.d.}}{\sim} \mathcal{N}(0, 1)$ with $n > 50$, and let $X_{(1)}, \cdots, X_{(n)}$ be the associated order statistics. Are there any references ...
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Finding Fisher information [duplicate]

Let $X$ distribution belongs for the family $\mathcal{P}\{P_{\theta}, \theta \in \Theta \}$. We need to find Fisher information $I(\theta)$ according $n$ simple sample, when $P_{\theta}$ is $N(\mu,\...
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extract distribution of y for point x given lm of y~x

If I have two variables x and y that have a linear relationship e.g. using data from the mtcars package and R code ...
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1answer
55 views

Interpreting qq-plot generated in R with a small sample of data

I have a small data sample with 10 firms and want to specify the distribution of their abnormal returns. First thing I did was to create the following histogram which indicates, when i am not ...
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0answers
10 views

Creating a vector of values to match a distribution

Let's say that I have a vector $u$ of real values (say of length 100). I know that $u-v$ is normally distributed with mean $\mu$ and variance $\sigma^2$. I would like to calculate a vector of ...
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0answers
6 views

consistency of non-linear squares estimators in Gaussian mixture model

Does any one know or have references for the consistency of non-linear squares estimators in Gaussian mixture model? (or the consistency of qausi mles in Gaussian mixture model). Thank you. I have ...