diff --git a/pres/counts.md b/pres/counts.md deleted file mode 100644 index 192cc11..0000000 --- a/pres/counts.md +++ /dev/null @@ -1,43 +0,0 @@ -# PDF - -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: -$$ - -$$ diff --git a/pres/moyal.md b/pres/moyal.md deleted file mode 100644 index aa16f13..0000000 --- a/pres/moyal.md +++ /dev/null @@ -1,35 +0,0 @@ -The Moyal distribution, which is a steepest descent approximation of the -Landau distribition, is defines as: -$$ - \exp \left( - \frac{x - \mu }{2 \sigma} - - \frac{1}{2} \exp \left( - \frac{x -\mu}{\sigma} \right) \right) -$$ -Mean $m$ and variance $\sigma$: -$$ - m = \mu + \sigma [ \gamma + \ln(2) ] \et \sigma = \frac{\pi^2 \sigma^2}{2} -$$ -Median: -$$ - \mu - \sigma \left[ 2 \text{erf}^{-1} \left( \frac{1}{2} \right)^2 \right] -$$ -skewness and kurtosis are constant: -$$ - s = \frac{28 \sqrt{2} Z(3)]{\pi^3} \et k = 7 -$$ -max value: -$$ - \frac{1}{\sqrt{2 e \pi}} -$$ -cdf: -$$ - \text{erf} \left( \frac{\exp \left( - - \frac{x - \mu}{2 \sigma} \right)}{\sqrt{2}} \right) -$$ - -$\mu$ is the location parameter and $\sigma$ is the scale parameter. -The Moyal distribution was first proposed in a 1955 paper by physicist J. E. -Moyal. The distribution models the energy lost by a fast charged particle -(and hence the number of ion pairs produced) during ionization. Historically, -the Moyal distribution has been utilized in the approximation of the Landau -Distribution and has since found use in modeling a wide array of phenomena. - diff --git a/pres/notes b/pres/notes deleted file mode 100644 index eec3ec2..0000000 --- a/pres/notes +++ /dev/null @@ -1,8 +0,0 @@ -produrre moyal più simile possibile alla Landau e poi distinguiamole. -renderle simili: - stessa moda - stessa fwhm - stessa mediana -poi applichiamo i quattro test che abbiamo implementato per distinguerle. - -La Moyal è come una gaussiana