Editing Dirac delta function (section)

====Probabilistic considerations====
In the context of [[probability theory]], it is natural to impose the additional condition that the initial {{math|''η''<sub>1</sub>}} in an approximation to the identity should be positive, as such a function then represents a [[probability distribution]].  Convolution with a probability distribution is sometimes favorable because it does not result in [[overshoot (signal)|overshoot]] or undershoot, as the output is a [[convex combination]] of the input values, and thus falls between the maximum and minimum of the input function.  Taking {{math|''η''<sub>1</sub>}} to be any probability distribution at all, and letting {{math|1=''η<sub>ε</sub>''(''x'') = ''η''<sub>1</sub>(''x''/''ε'')/''ε''}} as above will give rise to an approximation to the identity.  In general this converges more rapidly to a delta function if, in addition, {{mvar|η}} has mean {{math|0}} and has small higher moments. For instance, if {{math|''η''<sub>1</sub>}} is the [[uniform distribution (continuous)|uniform distribution]] on {{nowrap|1=<math display="inline">\left[-\frac{1}{2},\frac{1}{2}\right]</math>,}} also known as the [[rectangular function]], then:{{sfn|Saichev|Woyczyński|1997|loc=§1.1 The "delta function" as viewed by a physicist and an engineer, p. 3}}
<math display="block">
\eta_\varepsilon(x) = \frac{1}{\varepsilon}\operatorname{rect}\left(\frac{x}{\varepsilon}\right)=
\begin{cases}
\frac{1}{\varepsilon},&-\frac{\varepsilon}{2}<x<\frac{\varepsilon}{2}, \\
0,                    &\text{otherwise}.
\end{cases}</math>

Another example is with the [[Wigner semicircle distribution]]
<math display="block">\eta_\varepsilon(x)= \begin{cases}
\frac{2}{\pi \varepsilon^2}\sqrt{\varepsilon^2 - x^2}, & -\varepsilon < x < \varepsilon, \\
0, & \text{otherwise}.
\end{cases}</math>

This is continuous and compactly supported, but not a mollifier because it is not smooth.