Bell shaped function

From HandWiki
The Gaussian function is the archetypal example of a bell shaped function

A bell-shaped function or simply 'bell curve' is a mathematical function having a characteristic "bell"-shaped curve. These functions are typically continuous or smooth, asymptotically approach zero for large negative/positive x, and have a single, unimodal maximum at small x. Hence, the integral of a bell-shaped function is typically a sigmoid function. Bell shaped functions are also commonly symmetric.

Many common probability distribution functions are bell curves.

Some bell shaped functions, such as the Gaussian function and probability distribution of the Cauchy distribution, can be used to construct sequences of functions with decreasing variance that approach the Dirac delta distribution.[1] Indeed, the Dirac delta can roughly be thought of as a bell curve with variance tending to zero.

Some examples include:

[math]\displaystyle{ f(x) = a e^{-(x-b)^2/(2c^2)} }[/math]
[math]\displaystyle{ f(x) =\frac 1 {1+\left|\frac{x-c} a \right|^{2b}} }[/math]
[math]\displaystyle{ f(x) = \operatorname{sech}(x)=\frac{2}{e^x+e^{-x}} }[/math]
[math]\displaystyle{ f(x) = \frac{8a^3}{x^2+4a^2} }[/math]
[math]\displaystyle{ \varphi_b(x)=\begin{cases}\exp\frac{b^2}{x^2-b^2} & |x|\lt b, \\0 & |x|\geq b.\end{cases} }[/math]
[math]\displaystyle{ f(x;\mu,s) = \begin{cases} \frac 1 {2s} \left[ 1 +\cos\left(\frac{x-\mu}s \pi\right)\right] & \text{for }\mu-s \le x \le \mu+s, \\[3pt] 0 & \text{otherwise.} \end{cases} }[/math]
  • The derivative of the logistic function. This is a scaled version of the derivative of the hyperbolic tangent function.
[math]\displaystyle{ f(x)=\frac{e^x}{\left(1+e^x\right)^2} }[/math]
  • Some algebraic functions. For example
[math]\displaystyle{ f(x)=\frac{1}{(1+x^2)^{3/2}} }[/math]

Gallery

References