58 lines
778 B
Markdown
58 lines
778 B
Markdown
# Landau PDF
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## A pathological distribution
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Because of its fat tail:
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\begin{align*}
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E[x] &\longrightarrow + \infty \\
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V[x] &\longrightarrow + \infty
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\end{align*}
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. . .
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No closed form for parameters.
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## Landau median
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The median of a PDF is defined as:
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$$
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Q_L(m) = \frac{1}{2}
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$$
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. . .
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- CDF computed by numerical integration,
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- QDF computed by numerical root-finding (Brent)
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$$
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m_L = 1.3557804...
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$$
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## Landau mode
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- Maxmimum $\quad \Longrightarrow \quad \partial_x M(\mu) = 0$,
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- Computed by numerical minimization (Brent)
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$$
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\mu_L = − 0.22278...
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$$
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## Landau FWHM
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$$
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\text{FWHM} = x_+ - x_- \with L(x_{\pm})
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= \frac{L_{\text{max}}}{2} = \frac{L(\mu_L)}{2}
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$$
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- Computed numerically (Brent)
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$$
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\text{FWHM}_L = 4.018645...
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$$
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