Normal distribution

Creator

Creator

Created

Created

2020 Aug 23 13:6

Editor

Editor

Edited

Edited

2024 Nov 19 12:7

Refs

Refs

Gaussian Distribution, Univariate Gaussian

f(x) = e^{-x^2}, f(x) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{x^2}{2}}

Only 2 or 1 (

\sigma

) parameter can represent gaussian distribution. So we don’t loss resolution

Normal distribution 폐쇄성은

Probability Density Function 끼리 곱하거나 적분하거나

Convolution or Random variable 선형결합해도 가우시안이다. (not random variable for multiplication such as

chi Square Distribution)

If

\delta

approximate 0, the normal distribution goes to

$D$ is dimension of $x$

f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x - \mu)^2}{2 \sigma^2}}

N(\mu, \sigma^2) , N(\mu, \Sigma)

Normal Distributions

Multivariate Gaussian

Univariate Gaussian

Normal Distribution Notion

Standard Normal Distribution

Central Limit Theorem

Gaussian Identities

De Moivre - Laplace Theorem

Expectation

\mathbb{E}[X] = \int x f_{\mu, \sigma^2}(x) dx = \int y f_{0, \sigma^2}(y) dy + \mu = \mu.

Variance

\mathbb{V}(X) = \mathbb{E}[(X - \mathbb{E}[X])^2] = \mathbb{E}[X^2] - \mu^2.

Normal distribution

In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is The parameter is the mean or expectation of the distribution, while the parameter is the variance. The standard deviation of the distribution is (sigma). A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.

Normal distribution

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

Normal distribution

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