Statistics / Probability - Gaussian function (<math>e^{-x^2}</math>)

1 - About

In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distribution of complicated sums, according to the central limit theorem.

3 - Syntax

3.1 - Simplified

A Gaussian function is generally simplified by:

<MATH> f(x)=e^{\displaystyle -x^2} </MATH>


var exponent = (x) => Math.pow(x,2)
var func = (x) =>  Math.pow( Math.E ,  -exponent(x) );

// The graph
var graph = new world.func(func)
   .setTitle("The base gaussian function")
    .render();

3.2 - Detailled

A Gaussian is a function of the form:

<MATH> f(x)=ae^{-{\frac {\displaystyle(x-b)^{2}}{\displaystyle 2c^{2}}}} </MATH> where:

  • The variable a defines the position of its mode (ie peak) on the y axis.
  • The variable b defines the position of its mode (ie peak) on the x axis.
  • The variable c defines the spread

// the position of the mode (ie peak) on the ''y'' axis.
var a = 0.5; 
// The position of the mode on the x axis (ie the position of the pick on the x axis)
var b = -1; 
// The spread (the more the more spread)
var c = 1.5; 

// The gaussian function
var exponent = (x) => Math.pow((x-b),2) / (2*Math.pow(c,2))
var func = (x) =>  a * Math.pow( Math.E ,  -exponent(x) );

// The graph
var graph = new world.func(func)
   .setTitle("A gaussian function where a="+a+", b="+b+", c="+c)
    .render();

4 - Play


// the position of the mode (ie peak) on the ''y'' axis.
var a = 0.9; 
// The position of the modeon the x axis (ie the position of the pick on the x axis)
var b = -2; 
// The spread (the more the more spread)
var c = 2; 

// The gaussian function
var exponent = (x) => Math.pow((x-b),2) / (2*Math.pow(c,2))
var func = (x) =>  a * Math.pow( Math.E ,  -exponent(x) );

// The graph
var graph = new world.func(func)
   .setTitle("A gaussian function where a="+a+", b="+b+", c="+c)
    .render();

5 - Documentation / Reference


Data Science
Data Analysis
Statistics
Data Science
Linear Algebra Mathematics
Trigonometry

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