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Bell-shaped function
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<p>A bell-shaped function or simply 'bell curve' is a <a href="/facts/Function_(mathematics)/CdReEKCh">mathematical function</a> having a characteristic "<a href="/facts/Bell/oU1E1JHg">bell</a>"-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 <a href="/facts/Sigmoid_function/lAXu3AwW">sigmoid function</a>. Bell shaped functions are also commonly symmetric. </p><p>Many common probability distribution functions are bell curves. </p><p>Some bell shaped functions, such as the <a href="/facts/Gaussian_function/5bG8Kpp9">Gaussian function</a> and the probability distribution of the <a href="/facts/Cauchy_distribution/3HS6D3Xk">Cauchy distribution</a>, can be used to construct sequences of functions with decreasing <a href="/facts/Variance/ULBJKXD1">variance</a> that approach the <a href="/facts/Dirac_delta/V8PjRP8T">Dirac delta</a> distribution. Indeed, the Dirac delta can roughly be thought of as a bell curve with variance tending to zero. </p><p>Some examples include: </p> <ul><li><a href="/facts/Gaussian_function/5bG8Kpp9">Gaussian function</a>, the probability density function of the <a href="/facts/Normal_distribution/UapjjPyQ">normal distribution</a>. This is the archetypal bell shaped function and is frequently encountered in nature as a consequence of the <a href="/facts/Central_limit_theorem/0gc2L0Wd">central limit theorem</a>.</li></ul> f ( x ) = a e − ( x − b ) 2 / ( 2 c 2 ) {\displaystyle f(x)=ae^{-(x-b)^{2}/(2c^{2})}} <ul><li><a href="/facts/Fuzzy_logic/1PvN4Att">Fuzzy Logic</a> generalized membership bell-shaped function</li></ul> f ( x ) = 1 1 + | x − c a | 2 b {\displaystyle f(x)={\frac {1}{1+\left|{\frac {x-c}{a}}\right|^{2b}}}} <ul><li><a href="/facts/Hyperbolic_function/Y4Cb5S35">Hyperbolic secant</a>. This is also the derivative of the <a href="/facts/Gudermannian_function/bA0B2KcC">Gudermannian function</a>.</li></ul> f ( x ) = sech ( x ) = 2 e x + e − x {\displaystyle f(x)=\operatorname {sech} (x)={\frac {2}{e^{x}+e^{-x}}}} <ul><li><a href="/facts/Witch_of_Agnesi/hfSujFb9">Witch of Agnesi</a>, the <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> of the <a href="/facts/Cauchy_distribution/3HS6D3Xk">Cauchy distribution</a>. This is also a scaled version of the derivative of the <a href="/facts/Arctangent/bVwU9FPG">arctangent</a> function.</li></ul> f ( x ) = 8 a 3 x 2 + 4 a 2 {\displaystyle f(x)={\frac {8a^{3}}{x^{2}+4a^{2}}}} <ul><li><a href="/facts/Bump_function/ExdBtMHN">Bump function</a></li></ul> φ b ( x ) = { exp b 2 x 2 − b 2 | x | < b , 0 | x | ≥ b . {\displaystyle \varphi _{b}(x)={\begin{cases}\exp {\frac {b^{2}}{x^{2}-b^{2}}}&|x|<b,\\0&|x|\geq b.\end{cases}}} <ul><li>Raised cosines type like the <a href="/facts/Raised_cosine_distribution/uceLfxdC">raised cosine distribution</a> or the <a href="/facts/Raised-cosine_filter/QTE7SXr7">raised-cosine filter</a></li></ul> f ( x ; μ , s ) = { 1 2 s [ 1 + cos ( x − μ s π ) ] for μ − s ≤ x ≤ μ + s , 0 otherwise. {\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\leq x\leq \mu +s,\\[3pt]0&{\text{otherwise.}}\end{cases}}} <ul><li>Most of the <a href="/facts/Window_function/5Xj7h2ah">window functions</a> like the <a href="/facts/Kaiser_window/thJtvX74">Kaiser window</a></li></ul> <ul><li>The derivative of the <a href="/facts/Logistic_function/IaQ254t7">logistic function</a>. This is a scaled version of the derivative of the <a href="/facts/Hyperbolic_tangent_function/Y4Cb5S35">hyperbolic tangent function</a>.</li></ul> f ( x ) = e x ( 1 + e x ) 2 {\displaystyle f(x)={\frac {e^{x}}{\left(1+e^{x}\right)^{2}}}} <ul><li>Some <a href="/facts/Algebraic_functions/yT7EKmwF">algebraic functions</a>. For example</li></ul> f ( x ) = 1 ( 1 + x 2 ) 3 / 2 {\displaystyle f(x)={\frac {1}{(1+x^{2})^{3/2}}}}