# Bell-shaped function

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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. Indeed, the Dirac delta can roughly be thought of as a bell curve with variance tending to zero.

Some examples include:

$f(x)=ae^{-(x-b)^{2}/(2c^{2})}$ $f(x)={\frac {1}{1+\left|{\frac {x-c}{a}}\right|^{2b}}}$ $f(x)=\operatorname {sech} (x)={\frac {2}{e^{x}+e^{-x}}}$ $f(x)={\frac {8a^{3}}{x^{2}+4a^{2}}}$ $\varphi _{b}(x)={\begin{cases}\exp {\frac {b^{2}}{x^{2}-b^{2}}}&|x| $f(x;\mu ,s)={\begin{cases}{\frac {1}{2s}}\left[1+\cos \left({\frac {x-\mu }{s}}\pi \right)\right]&{\text{for }}\mu -s\leq x\leq \mu +s,\\[3pt]0&{\text{otherwise.}}\end{cases}}$ $f(x)={\frac {e^{x}}{\left(1+e^{x}\right)^{2}}}$ $f(x)={\frac {1}{(1+x^{2})^{3/2}}}$ 