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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continuous vs discrete random walk

For 1D random walk in discrete case the probability of finding walker at position X after N steps(P_N(X)) is binomial distribution(http://mathworld.wolfram.com/RandomWalk1-Dimensional.html), moreover ...
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45 views

sampling distribution for N(0,1) samples

Here is an portion of my lecture notes from class, we are studying sampling distributions. I am confused on some of the examples that are showed in the attached picture. For the first example, ...
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19 views

The mean of a class of students' weight is 120.3 pounds. The Standard Deviation is 15 [on hold]

The mean of a class of students' weight is 120.3 pounds. The Standard Deviation is 15. a.) Find the possibility of students' weight below 130 if 16 people are randomly selected. b.) construct a 95% ...
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24 views

Find Bayes rule/action under given prior

I am able to solve for Bayes actions/rules with no data and am able to follow problems with simple data. However, I'm not sure how to solve a question where the data, $X$, is conditional on the state ...
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9 views

If you transform response time data, e.g. to generate a CI, do you transform the values back for interpretation? [duplicate]

I would like to create a CI or highest density interval for response time data. The distribution of the response times is quite skewed and I think about transforming them by LN(y). However, my ...
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21 views

Expected value of normal-wishart, is the solution correct?

I want to compute expected value of $E[μΛ]$ for a normal-wishart distribution how can i compute it? A normal-wishart distribution is defined as below: $$ NW(μ,Λ|μ_0,λ,W,v)=N(μ|μ_0,(λΛ)^{−1})W(Λ|W,v) ...
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11 views

Residual Not Normal for Model Seasonal Time Series in R

I got a problem when choose the model for forecasting with time series. I'm in a middle writing my Thesis. My data have a seasonal pattern so, i tried use this model ...
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1answer
25 views

Shapiro-Wilk normality test - how to interpret this?

So, I'm having data which represent two groups, one that used tool 1 and the other that used tool 2. I asked a series of questions (10 questions) and I recorded the time as well. Afterwards I ...
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19 views

integral of trace function [migrated]

How can I compute the below integral? $$ \int e^{-1/2*tr[(\mu\mu^T-\mu m^T-m\mu^T)\Sigma]}d\mu $$ in which $\mu \in R^{n},m \in R^{n},\Sigma \in R^{n*n}$ and $tr(.)$ is trace of matrix. I have ...
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16 views

Expectation of dependent variables in normal-Wishart distribution

I want to compute $E[\mu\Lambda]$ for a normal-Wishart distribution how can I compute it? A normal-wishart distribution is defined as below: ...
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15 views

Composition of Normals

I.e., the data was generated from 5 normal distributions: ...
3
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1answer
27 views

Normal Probability Plot of Residuals

Below is a normal probability plot of residuals from my lecture The NSCORE(z score) is quite confusing. For example, the first nscore is -1.54664, which should be 0.061 or 61% percentile, it doesn't ...
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0answers
29 views

Explanation of statistic seen on TV (related to elections) [duplicate]

I saw the following statistic on TV a few years back: 53% of voters are going to vote for Mitt Romney over Barack Obama (Error: 3%, sample size: 300, survey conducted via phone) After seeing ...
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0answers
13 views

Testing for the ordering of several means

We have delineated 4 groups into a population and have a theory according to which the distributions of a certain characteristic of this popular are ordered in the sense that $\mu_i > \mu_{i+1}$ ...
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1answer
28 views

How to get the determinant of a covariance matrix from its diagonal elements

I am trying to implement a speaker recognition system in MATLAB. I am using Gaussian Mixture Models (GMM) for speaker modelling and maximizing the posterior probabilities for classification. The ...
2
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0answers
17 views

DFA in SPSS: Sorts effectively, but Box's M is still 0.000. Is the analysis worthless?

I am a geologist attempting to apply the discriminant function analysis to surface features I have mapped in ArcGIS. At the moment I have 4 dimensionless sorting variables calculated for each feature, ...
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2answers
51 views

Finding the conditional distribution of 2 dependent normal random variables

Here's the situation $X \sim N(\mu, \sigma^2)$ and given $X=x$, $Y \sim N(x, \tau^2)$ I need to find the distribution of $X$ given $Y=y$ From what's given, I know the pdf's of $X$ as well as ...
4
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1answer
97 views

Distribution of $|q|^2$ if $\Re[q]\sim \mathcal{N}(\mu_1,\sigma^2/2)$ and $\Im[q]\sim \mathcal{N}(\mu_2,\sigma^2/2)$

Let $q$ be a complex random variable such that: $\Re[q]\sim \mathcal{N}(\mu_1,\sigma^2/2)$ and $\Im[q]\sim \mathcal{N}(\mu_2,\sigma^2/2)$. What is the PDF and CDF of the squared norm $|q|^2$ ?
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1answer
51 views

Pdf of the square of a standard normal random variable [closed]

I have this problem where I must find the pdf of $Y = X^2$. All I know is that $X$ has the distribution $N(0,1)$. What kind of distribution is $Y = X^2$? Same as $X$? How do I find the pdf?
6
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44 views

References that justify use of Gaussian Mixtures

Gaussian mixture models (GMMs) are appealing because they are simple to work with both in analytically and in practice, and are capable of modeling some exotic distributions without too much ...
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27 views

Bias of a Gaussian

Given a set of observations $X=(x_1,\ldots,x_n), x_i \in \mathbb{R}^d$, the maximum-likelihood Gaussian $\mathcal{N}(\mu,\Sigma)$ is given by $\mu = \bar{x}$ and $\Sigma = \widehat{\sigma}^2$, i.e., ...
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2answers
66 views

Customization of a standard Bell Curve

Hopefully this isn't a duplicate, I've tried to search for similar things, but to no luck. I'm curious on how you would computationally compute a random distribution of numbers that follows a bell ...
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0answers
13 views

random data distribution

I have data with no trivial probability distribution function (kind of random data). I want to feature the data distribution as a sum of commun probability density functions (a sum of gaussian or ...
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1answer
122 views

Marginal normality and joint normality

Let $X$ and $Y$ be two independent standard normally distributed random variables $N(0,1)$ .If we define a new random variable $Z$ such that : $$Z = \begin{cases}X & \text{if} &XY > ...
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segments of normal distribution are normally distributed?

I need a hint how to prove following: Log fold changes follow normal distribution. On the plot you can see log2 fold changes versus mean. If I segment log2 fold changes into the bins, so that I have ...
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1answer
12 views

Scaling the probability mass function

I have produced a histogram of the frequency of observing some variable, call it $x$. I have then used the following equation: $$f(x;\mu,\sigma)=\frac{1}{\sqrt{2\pi ...
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14 views

Testing for normality - low p-value

I've a dataset with about 12k values given. It looks like this: When I try to test for normality I get following results, where p-value is extremely slow: ...
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1answer
17 views

combination of normal distribution samples

According to sums of Independent Normal Random Variables Let's say we have two samples. Sample 1 follows a N(u1, var1) and Sample 2 follows a N(u2, var2), Case 1: I take one subject from each ...
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2answers
110 views

Rate at which a Gaussian random variable is the maximum in a set of independent Gaussian random variables [duplicate]

Assume a random vector $X = [x_1, x_2, ..., x_n]$ where the random variables are independent and Gaussian distributed. At the same time, they are not identically distributed; they can have arbitrary ...
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2answers
170 views

What is the distribution of $e=Y-\mathbb{E}(Y)$ where $Y=\exp(u), \ \ \ u\sim\mathbb{N}\left(\mu,\sigma^2\right)$

As $Y$ is log-normal we've $Y\sim \mathbb{LN}\big(\exp(\mu+\sigma^2/2),\exp(2\mu+\sigma)(\exp(\mu^2)-1)\big)$. Now I define $e = Y - \mathbb{E}(Y) = Y - \exp(\mu+\sigma^2/2)$. As $e$ is the ...
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32 views

Data Fitting using R Code

I'm trying to use a copula function but I do not have the equation for my data, I just have all my raw data. Since my data is not normal, I tried fitting it into a normal distribution using the ...
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23 views

Ratio of Fourier Transform Gaussians (in Matlab)

this is more of a theoretical question as the implementation doesn't really matter. I'm experimenting in Matlab, and I was curious about something. I know that the division of Gaussian-distributed ...
3
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1answer
20 views

Deconvolution of two Gaussians

Assuming $X$ and $Y$ are two Gaussians with parameters of $\mu_X,\Sigma_X$ and $\mu_Y,\Sigma_Y$ then for their convolution we know that (reference) : $Z=X*Y$ is also a Gaussian with parameters of ...
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3 views

Is linearity an issue for structural learners?

I have a matrix of z-scores. Let's say these z-scores are trustworthy and that the assumption that the data fits a normal distribution has been tested for our data. I have a large feature space ...
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1answer
28 views

Normal approximation of parameter $p$ of $Bin(n,p)$? [closed]

I've seen normal approximation applied for approximating a binomial distribution $B(n,p)$. However, if one estimates the parameter $p$, then can the parameter $p$ be "normally approximated" just as ...
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2answers
41 views

Variance of Gaussian linear combination

I have two independent gaussian distributions to combine and I have a doubt. Let's say we have $X \sim N(\mu_x,\sigma^2_x)$ and $Y \sim N(\mu_y,\sigma^2_y)$. I want to mix the two variables with a ...
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2answers
32 views

How to calculate Standard error of Normal Distribution with given Confidence Intervale

I have one basic question in preforming the forecast of fertility rate. I assumed if the fertility rate (is normally distributed) is equal to 3.0, there is an 80 per cent chance that fertility would ...
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2answers
71 views

How to find the variance of $C\hat\beta$?

Assuming a Gauss-Markov model such that $H_o$: $C\beta$ = $d$, how do I prove that the variance of $C\hat\beta$~ $N(C\beta, \sigma^2C(x'X)^-C')$ ? My Work...Which I Know is Not Correct, when ...
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31 views

Calculating the marginal likelihood with multiple observations in a multivariate Normal-Normal model

Given $f, y_1, \ldots, y_n \in \mathcal{R}^d$ and $V$ fixed: $f \sim N(f; 0, V)$ $y_i | f \sim N(y_i; f, \sigma^2I_d)$ for $i = 1, \ldots, n$ [so they're iid] Find the marginal likelihood: $p(y_1, ...
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11 views

Matrix is singular warning, RCOND = NaN in the EM-step of GMM

In the EM step of GMM, I call a function gaussianND as: pdf(:, j) = gaussianND(unseen_data, mu(j, :), sigma{j}); which evaluates gaussian for all data points ...
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1answer
16 views

Do I have to test normality for both groups when comparing from a single population?

This should be an easy one. I'm a novice when it comes to statistics and English isn't my first language so bear with me. I have one population that numbers about 700. Of these 700, 25 are of special ...
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11 views

How to find density function of (x+|x|)/2

when X follows N(0,sigma^2).what is the density function of (x+|x|)/2
3
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54 views

Compound Distribution — Uniform Distribution with Normally Distributed Parameters

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Uniform Distribution whose parameters are distributed ...
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1answer
31 views

distribution of sample variance of correlated observations

It is well known that if we have n i.i.d. observations of a normal random variable, then Cochran's theorem tells us that: $\frac{(n-1)s^2}{\sigma^2} \widetilde{} χ^2_{n-1}$ But what if the samples ...
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15 views

Reducing the co-variance of data for denoising, with minimum change in the mean

Assume $X$ and $Y$ are two sets of $n$ dimensional feature vectors from two different multivariate Gaussian distributions with the covariance of $\Sigma_X$ , $\Sigma_Y$ and the mean of $\mu_X$, ...
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1answer
66 views

Can my data be assumed to be normally distributed?

I have read a lot about when to use a (paired) t-test or a Wilcoxon Signed Rank Test as the non-parametric alternative but I need your help: I have gathered some paired data and want to perform some ...
3
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1answer
126 views

Distribution of variance of Gaussian variable

I have a Gaussian random variable, which I can use to generate a sequence of values. So, I've generated a sequence of values of arbitrary length, and each set of 50 data become a sample. Now, consider ...
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9 views

Express Normal Distributions with Integral Notation?

If possible, how would I be able to express some Normal Distribution using Integral notation if I had a distribution such that $$ \mu = 188 \text{ units} $$ $$ \sigma = 41 \text{ units} $$ And we ...
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0answers
51 views

Finding what happened between two PDFs from the parameters of a resulting PDF

Assuming $X$ and $Y$ are Gaussian random variables with PDFs of $f(X)$ and $g(Y)$ with parameters of $(\mu_x, \Sigma_x)$ and $(\mu_y, \Sigma_y)$, we know that: for the operation of sum ($+$), if ...
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1answer
42 views

Why does probabilistic PCA use Gaussian prior over latent variables?

I am currently reading papers about probabilistic PCA and I am wondering why is Gaussian prior (and not some other prior) chosen for the latent variables? Is it just because it's simple or is there ...