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Probability Integral

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$\displaystyle \alpha(x)$ $\textstyle \equiv$ $\displaystyle {1\over\sqrt{2\pi}} \int^x_{-x} e^{-t^2/2}\,dt$ (1)
  $\textstyle =$ $\displaystyle \sqrt{2\over\pi} \int_0^x e^{-t^2/2}\,dt$ (2)
  $\textstyle =$ $\displaystyle 2\Phi(x)$ (3)
  $\textstyle =$ $\displaystyle \mathop{\rm erf}\nolimits \left({x\over\sqrt{2}}\right),$ (4)

where $\Phi(x)$ is the Normal Distribution Function and Erf is the error function.

See also Erf, Normal Distribution Function




© 1996-9 Eric W. Weisstein
1999-05-26