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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9 views

What distribution is appropriate for length of a signal (in days)?

I have a vector of "signal lengths", which is the amount of days a signal lasts once initially activated. For the purpose of confidence intervals and hypothesis testing, do I want to use a Poisson ...
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0answers
9 views

The probability that several normally distributed random variables have a particular order

Let's say we have n random, independent variables $X_1,\ldots,X_n$ with normal distributions. Is there a reasonable way how to compute the probability that they have a particular order, for example ...
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3answers
343 views

Probability that the minimum of a normal random sample will exceed the maximum of another?

I sample independently $n$ data points following normal distribution with $\mu = 0$ and $\sigma = 1$. Then I divide the sample into two groups $G_1$ and $G_2$ of sizes $g_1$ and $g_2$ respectively ...
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0answers
29 views

Non normal and nonlinear data

I'm working with pilot study data (n=70) that has two continuous response variables and a number of categorical and continuous predictors. Naturally my data does not follow a normal distribution nor ...
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1answer
26 views

Half-normal distribution and correlation?

In health sciences many variables may exhibit half-normal distribution. For example an inflammation process regarding one cell type or population in a tissue sample. The lack of the inflammatory cells ...
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0answers
26 views

probability distribution of complex Gaussian column vector and conditional probability of complex Gaussian column vector

I have column vector $\vec r=[r_1\ r_2]^T$. $$\vec r =hA\vec s +\vec n$$ where $h$ is a complex number , $\vec n \sim \mathcal{C} \mathcal{N}(\begin{bmatrix} 0 \\ 0 ...
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0answers
21 views

Correlation preserving transformation conundrum

I have a problem where I need to generate $n$ random variables $\in$ [0,1] (you can think of them as some sort of probabilities) and the variables have a known correlation structure given by a ...
2
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1answer
35 views

Multivariate normal distribution

I have 4 r.v $(X,Y,W,Z)$ distributed as a multivariate normal. Mean and variance-covariance matrix are known. Is it possible to calculate, for example $\mathrm{Prob} (aX+bY < k, W >0,Z ...
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0answers
23 views

Lorentz and Gaussian

I would like to ask the difference between Lorentz and Gaussian. In laser application, the intensity of Natural and Doppler broadening is different in a mathematical way simply due to the differences ...
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0answers
14 views

How to mathematically prove that the continuous data is better for finding out the correlations than the binary data?

If I want to calculate the correlations among the components in a vector space using the MLE with a prior of multivariate Normal distribution, which kind of data should be better? the binary data or ...
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1answer
43 views

What is the distribution of the absolute difference between two points sampled from a normal distribution?

I have a normally distributed random variable $X$. I sample two points $x_1$ and $x_2$, and I am interested in the absolute difference between these two sampled points: $d=|x_2-x_1|$. I repeat this ...
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0answers
26 views

Gibbs sampling version for estimating Hierarchical Double Dirichlet Process Mixture of Gaussian Processes

I'm trying to implement Gibbs sampling to estimate the parameters of the following non-parametric model: $$\begin{align*} \beta|\gamma & \sim \text{GEM}(\gamma)\\ k_t|\beta & \sim \beta\\ ...
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1answer
46 views

The squared-norm of the projection of a Gaussian vector onto an independent $d$-dimensional subspace is a $\chi^2_{2d}$

How we can prove that: The squared-norm of the projection of a $N$-dimensional complex vector with i.i.d. unit-variance and zero mean Gaussian components onto an independent $d$-dimensional subspace ...
2
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1answer
46 views

If $X$ and $Y$ are normally distributed random variables, what kind of distribution their sum follows?

I was reading this question. It is about notation but I would like to ask something regarding the sum of two normally distributed random variables. If $X$ is a normally distributed random variable ...
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1answer
28 views

Should propensity distributions be normally distributed?

When you use a model to predict the propensity of customers responding to a campaign. Should the distribution of the propensities be normally distributed or skewed to one side?
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1answer
59 views

About the $X\sim \mathcal{N}$ notation

If a random variable $X$ has mean $\mu_{X}$ and variance $\sigma_{X}^{2}$, and follows a normal distribution, it may be written as $X\sim \mathcal{N}(\mu_{X},\sigma_{X}^{2})$. Suppose $Y$ is also ...
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0answers
9 views

Instances of this Gaussian variance model with non-convex log-likelihood

On page 4 of Thomas Minka's Beyond Newton's Method, he mentions the following Gaussian variance model with a non-convex maximum likelihood objective. What are some examples/instances of models where ...
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1answer
20 views

How to find the probability distribution of a two-variable metric?

I have a certain metric $f(x,y) \in [-1, 1]$. I would like to study the metric assuming $x$ and $y$ are random variables. The input variables are typically normally distributed in most applications of ...
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15 views

Given two normally distributed random variables, how can I find the largest standard deviation so that $p<\alpha$ using Student's t-test? [closed]

Suppose $X\sim N(\mu_{X}, \sigma^{2})$ and $Y\sim N(\mu_{Y}, \sigma^{2})$. If $\mu_{X}$ and $\mu_{Y}$ are known, how can I find the largest pooled standard deviation for the Student's t-test so that ...
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2answers
31 views

How to empirically show that a certain quantity approximately follows a normal distribution?

To motivate some theoretical work, I need to show that two certain variables (say $X$ and $Y$) approximately follow a normal distribution in actual datasets. I have one large dataset with about $300$ ...
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1answer
35 views

CLT simulation with R

I am trying to run a simple CLT simulation with R. I want to create a vector with 10000 dice rolls, then take 100 means of samples of "n" size to see how when "n" increases, the distribution starts ...
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20 views

Strictly positive random variables

Suppose $X\sim N(\mu, \sigma^{2})$ with some small $\sigma^{2}$ and largish $\mu$. Now $X$ will be rarely negative. Suppose I need random variables that are strictly positive but otherwise ...
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39 views

Standard deviation of sampling distribution of mean

If we take a sample and calculate the mean, we can calculate the standard deviation for the sampling distribution of the mean using this formula: $\sigma / \sqrt{n}$ But, how many samples do we need ...
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0answers
13 views

why use weighted sum of component densities in unsupervised parametric estimation?

In unsupervised parametric estimation why do we take $p(x|\Theta)$ as $ \sum_j P(w_j)p(x|w_j ,\Theta_j ), j = 1 - c$? That is weighted sum of component densities. where $p(x|\Theta)$ is mixture ...
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41 views

Peak heavy data filter

I am trying to replicate the motion of a measurement needle with a set radius crossing over a rough surface in order to measure the surface texture. How can I create a filter to replicate this motion ...
5
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0answers
44 views

why use diagonal $\Sigma$ when working with Bayes decision theory?

My prof. said in the class that for Bayes decision rule, the likelihood is Gaussian and in practice, we will almost always work with a diagonal $\Sigma$. Why is that? I know that a diagonal $\Sigma$ ...
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1answer
32 views

pattern classification when the prior probabilities are not equal

In the case of 2 class classification, the decision boundary occurs when 2 discriminant functions are equal: $$ g_1(x) = g_2(x) $$ $$ g_i(x) = p(x|w_i)P(w_i) $$ $$ p(x|w_i) = ...
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6answers
259 views

How is the notation $X\sim N(\mu,\sigma^2)$ read?

How is the notation $X\sim N(\mu,\sigma^2)$ read? Is it $X$ follows a normal distribution? Or $X$ is a normal distribution? Or perhaps $X$ is approximately normal.. What if there are several ...
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1answer
43 views

Determining if two correlated Gaussian RVs are jointly Gaussian

One common way we can find the posterior distribution of Gaussian parameters for a Kalman filter on Gaussian observations is by first computing the covariance between the parameters and forecast given ...
3
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1answer
52 views

About the central limit theorem and statistical testing

Wikipedia states that In probability theory, the central limit theorem (CLT) states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent ...
5
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3answers
92 views

How to interpret this QQ plot?

I have ran a QQ plot on R on my data using par(mfrow=c(1,2)) par(pty="s") qqnorm(TEDS$LST1); qqline(TEDS$LST1) which gave me this: The histogram of the data ...
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0answers
28 views

click through rate stats

I have a group of 10 users. Each user receives a daily notification for one week. Ideally, each user will click on all notifications received. At the end of the week I collect the number of clicks per ...
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1answer
39 views

Variance of the $\hat{\sigma^2}$ of a Maximum Likelihood estimator

Given some normally distributed observations $x_1,x_2,...,x_n$ $\forall i\ x_i\sim\mathcal{N}(\mu, \sigma^2)$ the ML estimator decides that the variance that maximizes the likelihood function is ...
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1answer
25 views

Testing the difference between two samples with very different sizes using an un-paired t-test

I want to test wether population A is significantly different from population B (at 5% confidence interval). I need to do it for each variable using an un-paired t-test, since my data are normally ...
2
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1answer
27 views

Conceptual proof that conditional of a multivariate Gaussian is multivariate Gaussian

I understand the arithmetic derivation of the PDF of a conditional distribution of a multivariate Gaussian, as explained here, for example. Does anyone know of a more conceptual (perhaps, co-ordinate ...
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1answer
18 views

About using the standard normal table

Suppose $\alpha=0.05$. How do I find the value $z_{\alpha/2}$ from a standard normal table? So $\alpha/2 = 0.025$. If I look the values $z=0.02$ and $z=0.03$ they are $0.5080$ and $0.5120$. I don't ...
2
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1answer
38 views

Why does phase randomizing the Fourier transform of a data set render it Gaussian?

Let's say I have a data set $s_n$. I take the Fourier transform of this data set to obtain $\tilde{s}_n$. I randomize the complex phases of $\tilde{s}_n$ and I take an inverse transform to obtain ...
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1answer
57 views

Probability of sum of independent normally distributed variables [closed]

The weight of an adult swan is normally distributed with a mean of 30 pounds and a standard deviation of 9.8 pounds. A farmer randomly selected 36 swans and loaded them into his truck. What is the ...
2
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0answers
25 views

Assuming normality in RKHS

There are some authors who assume data in a Reproducing Kernel Hilbert Space is following a normal distribution. For example, in this article, the authors use this assumption to be able to derive ...
3
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0answers
56 views

What is the proof of $\mathbb{E}\Phi (X) = \Phi\left(\frac{\mu}{\sqrt{1+\sigma^2}}\right)$, where $X \sim \mathcal{N}(\mu,\sigma^2)$? [duplicate]

Let $X \sim \mathcal{N}(\mu,\sigma^2)$. I think it's true that $$\mathbb E \Phi(X) = \Phi \left(\frac{\mu}{\sqrt{1+\sigma^2}}\right)$$ where $\Phi$ is the cdf of standard normal. This holds up under ...
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0answers
34 views

R - Multivariate Gaussian Outlier Detection

I have a dataset with N samples (>200) and 5 variables. In my implementation I need to classify some samples after a "calibration". This calibration is done by using an amount of the dataset (i.e. 100 ...
5
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1answer
182 views

A closed form formula for the normalizing constant in standard normal auto-regressive series?

Let $Z_t = c_1Z_{t-1} + c_2Z_{t-2} + ... + c_nZ_{t-n} + c\epsilon_t$ where $Z_t, \epsilon_t \sim \mathtt{N}(0,1)$ are iid variables and $Z_s \sim \mathtt{N}(0,1)$ for all $s$. Given the values of ...
1
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2answers
79 views

Why does the variance decrease when averaging zero-mean gaussians?

I have the following in my textbook: $$ r_k \thicksim N(0,\sigma_r^2) \\ \Rightarrow \sum\limits_{k=1}^K r_k(t) \thicksim N(0,K\sigma_r^2) \\ \Rightarrow \frac1{K} \sum\limits_{k=1}^K r_k(t) ...
6
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1answer
110 views

How to calculate probability of observing a value given a permutation distribution?

I have a single observation with value $x = 0.5$ that comes out from a complicated computational process. I would like to know what is the probability to observe such value by chance. To attempt to ...
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0answers
46 views

4 cases of Maximum Likelihood Estimation of Gaussian distribution parameters

Let $x_1,x_2,...,x_n$ some normally distributed observations. So $\vec{x}=\begin{bmatrix}x_1 & x_2 & ... & x_n\end{bmatrix}^{T}$ In the context of my research I am trying to estimate ...
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0answers
26 views

What is the probability of at least one of a sum of normally distributed variables being greater than a threshold?

Let's say I have a pencil factory. I don't mind if my pencils are too short, but it's important they're not too long. My Quality Control department measure each pencil, and their measurement will be ...
3
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1answer
100 views

Transform normal distribution to skewed distribution without changing its support

I've found many questions and answers about transforming skewed distribution to normal. This question might arise because the simplicity of working with normal data. But, is there any function that ...
0
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0answers
56 views

How do I solve a under-determined quadratic multi-variate system?

I performed some simulations with known values of input variables $X_1$, $X_2$ and $X_3$, to find output response $Y$. The variables are distributed as following: $$ X_1 = N(\mu_1, \sigma_1) $$ $$ ...
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1answer
61 views

Does p value below 0.05 in mvShapiro.Test mean multivariate normality or not?

I am performing mvShapiro.Test from mvShapiroTest package. I get ...
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1answer
37 views

Normal distribution in R: Find the value of $x$ for $P(X=x) = c$ where $c$ is known

I have a normal distribution with parameters $\mu=750$, $\sigma=260$. I'm interested in finding the value of $x$ that satisfies $P(X=x)=0.001$ for both sides of the tail. How would I go about doing ...