analistica/pres/counts.md

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The Moyal distribution is defined as:

  M(x) = \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2} \left[ x + e^{-x} \right]}

More generally, it is defined with the location and scale parameters \mu and \sigma such as:

  x \rightarrow \frac{x - \mu}{\sigma}

CDF

The cumulative distribution function \mathscr{M}(x) can be derived from the pdf M(x) integrating:

  \mathscr{M}(x) = \frac{1}{\sqrt{2 \pi}} \int\limits_{- \infty}^x dy \, M(y)
  = \frac{1}{\sqrt{2 \pi}} \int\limits_{- \infty}^x dy \, e^{- \frac{1}{2}}
    e^{- \frac{1}{2} e^{-y}}

with the change of variable: \begin{align} z = \frac{1}{\sqrt{2}} e^{-\frac{y}{2}} &\thus \frac{dz}{dy} = \frac{-1}{2 \sqrt{2}} e^{-\frac{y}{2}} \ &\thus dy = -2 \sqrt{2} e^{\frac{y}{2}} dz \end{align} hence, the limits of the integral become: \begin{align} y \rightarrow - \infty &\thus z \rightarrow + \infty \ y = x &\thus z = \\frac{1}{\sqrt{2}} e^{-\frac{x}{2}} = f(x) \end{align} and the CDF can be rewritten as:

  \mathscr{M}(x) = \frac{1}{2 \pi} \int\limits_{+ \infty}^{f(x)}
  dz \, (- 2 \sqrt{2}) e^{\frac{y}{2}} e^{- \frac{y}{2}} e^{- z^2}
  = \frac{-2 \sqrt{2}}{\sqrt{2 \pi}} \int\limits_{+ \infty}^{f(x)}
  dz e^{- z^2}

since the erf is defines as: