Questions tagged [normal-distribution]

This tag is for questions on the Gaussian, or normal probability distribution, which may include multi-dimensional normal distribution. The normal distribution is the most common type of distribution assumed in technical stock market analysis and in other types of statistical analyses.

Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

Graphically, normal distribution will appear as a bell curve.

enter image description here

However, many other distributions are bell-shaped (such as the Cauchy, Student's $~t-~$, and logistic distributions).

Probability density function: The general formula for the probability density function of the normal distribution is \begin{equation*} P(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-~\frac{(x-\mu)^2}{(2\sigma^2)}},~x\in (-\infty,\infty). \end{equation*}

where

  • $~\mu~$ is the mean or expectation of the distribution (and also its median and mode),
  • $~\sigma~$ is the standard deviation, and
  • $~\sigma ^{2}~$ is the variance.

The case where $~μ = 0~$ and $~σ = 1~$ is called the standard normal distribution. The equation for the standard normal distribution is

\begin{equation*} P(x)=\frac{1}{\sqrt{2\pi}}e^{-~\frac{x^2}{2}},~x\in (-\infty,\infty). \end{equation*}

Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function.

Applications: The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, they become normally distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal. Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.

References:

https://en.wikipedia.org/wiki/Normal_distribution

http://mathworld.wolfram.com/NormalDistribution.html

6655 questions
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Large powers of sine appear Gaussian -- why?

As part of approximating an integral, I have noticed that $\sin^k(x), x\in[0, \pi]$ look almost identical to $\exp\left(-\frac{k}{2}(x-\frac{\pi}{2})^2\right)$ once $k$ is large enough (in practice, the two equations are visually identical for…
Søren Hauberg
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Scaling the normal distribution?

I might just be slow (or too drunk), but I'm seeing a conflict in the equations for adding two normals and scaling a normal. According to page 2 of this, if $X_1 \sim N(\mu_1,\sigma_1^2)$ and $X_2 \sim N(\mu_2,\sigma_2^2)$, then $X_1 + X_2 \sim…
alecbz
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Derivation of the density function of student t-distribution from this big integral.

My lecturer posed a question where we derive the density function of the student t-distribution from the Chi-square and Standard normal distribution. I worked on this question for days, and I am pretty sure the below integral is correct (Verified by…
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'normally distributed random numbers' vs 'uniformly distributed random number'?

what is the difference between 'normally distributed random numbers' and 'uniformly distributed random number'? A answer in a layman language is appreciated :)
user3130920
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Closed-form analytical solutions to Optimal Transport/Wasserstein distance

Kuang and Tabak (2017) mentions that: "closed-form solutions of the multidimensional optimal transport problems are relatively rare, a number of numerical algorithms have been proposed." I'm wondering if there are some resources (lecture notes,…
23
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How to calculate the integral in normal distribution?

The factory is making products with this normal distribution: $\mathcal{N}(0, 25)$. What should be the maximum error accepted with the probability of 0.90? [Result is 8.225 millimetre] How will I calculate it? How to integrate: $\exp\left(-…
user1111261
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What is the expectation of $ X^2$ where $ X$ is distributed normally?

I know that if $X$ were distributed as a standard normal, then $X^2$ would be distributed as chi-squared, and hence have expectation $1$, but I'm not sure about for a general normal. Thanks
maliky0_o
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Uniform distribution on the surface of unit sphere

It is known that given $X=(X_1, X_2, \ldots, X_n)$ iid $\sim N(0,1)$, then $X/\sqrt{X_1^2+\cdots+X_n^2}$ is uniformly distributed on the surface of unit sphere. Intuitively, I know that that's because the probability of $X/\sqrt{X_1^2+\cdots+X_n^2}$…
Julie
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Calculation of the Covariance of Gaussian Mixtures

I have a Gaussian mixture model, given by: $$ X \sim \sum_{i = 1}^M \alpha_i N_p(\mu_i, C_i) $$ such that $\sum_{i=1}^M\alpha_i =1 $. Is there a way I can compute the overall covariance matrix if $x$? I would like to say "$X$ has a covariance matrix…
NSR
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Average norm of a N-dimensional vector given by a normal distribution

I'm interested in knowing what is the expected value of the norm of a vector obtained from a gaussian distribution in function of the number of dimensions $N$ and $\sigma$, i.e: $$E[\|x\|_2],\quad x\sim\mathcal{N}(0,\sigma I_N)$$ I tried to search…
jmacedo
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Expected value of maximum and minimum of $n$ normal random variables

Let $X_1, \dots, X_n \sim N(\mu,\sigma)$ be normal random variables. Find the expected value of random variables $\max_i(X_i)$ and $\min_i(X_i)$. The sad truth is I don't have any good idea how to start and I'll be glad for a hint.
user65985
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P.d.f of the absolute value of a normally distributed variable

I came across this question as an exercise, had a brief idea, but didn't know how to proceed. Let $X \sim N(0, 1)$. What is the p.d.f of $|X|$ ? I know the final p.d.f looks just like the right half of the original pdf, but extended vertically for…
Sam Shen
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If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$

Let $A$ be a positive definite real $n\times n$ matrix. How can I prove that $$ \int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\,\det\left(\pi^{-1}{A}\right)\right|^{-1/2}=\pi^{n/2}\lvert\,\det A\rvert^{-1/2}\!, $$ where…
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length of Gaussian Random Vector

Suppose I have a random vector $x=[x_1,...,x_k]$ s.t. $x∼N(\mu,\sum)$. How is the length or magnitude of $x$ distributed? I know that if $k=2$ and $\sigma_1=\sigma_2$ and $\sigma_{12}=0$ ($x_1$ and $x_2$ are not correlated), it is Rayleigh…
simomo
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Is this distribution already known and has a name?

My question is whether the distribution on $\Bbb R$ with probability density $$ f(x) := \frac 2 {\sqrt{2\pi}} e^{-\frac{x^2}{2}} - 2 \vert x\vert \int_{\vert x \vert}^\infty \frac 1{\sqrt{2\pi}} e^{-\frac{y^2} 2} \text d y$$ is already appearing in…