section_path: Applications/Standard normal distribution
section_name: Standard normal distribution
paragraph_text: Standard normal distribution
  
  The normal distribution with zero mean and unit standard deviation is known as the standard normal distribution, given by the probability density function ϕ (x) = 1 2 π e − 1 2 x 2. {\displaystyle \phi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}.}
  
  The constraint of unit standard deviation (and thus also unit variance) results in the 1 / 2 in the exponent, and the constraint of unit total area under the curve ϕ (x) {\displaystyle \phi (x)} results in the factor 1 / 2 π {\displaystyle \textstyle 1/{\sqrt {2\pi }}}. This function is symmetric around x = 0, where it attains its maximum value 1 / 2 π {\displaystyle \textstyle 1/{\sqrt {2\pi }}}, and has inflection points at x = ±1.