analistica/slides/sections/2.md

Landau PDF

A pathological distribution

Because of its fat tail:

\begin{align*} E[x] &\longrightarrow + \infty \ V[x] &\longrightarrow + \infty \end{align*}

. . .

No closed form for parameters \thus numerical estimations

Landau median

The median of a PDF is defined as:

  m = Q \left( \frac{1}{2} \right)

. . .

• CDF computed by numerical integration
• QDF computed by numerical root-finding (Brent)

  m_L\ex = 1.3557804...

Landau mode

• Maximum \quad \Longrightarrow \quad \partial_x L(\mu) = 0

. . .

• Computed by numerical minimization (Brent)

  \mu_L\ex = − 0.22278...

Landau FWHM

We need to compute the maximum:

  L_{\text{max}} = L(\mu_L)

  \text{FWHM} = w = x_+ - x_- \with L(x_{\pm}) = \frac{L_{\text{max}}}{2}

. . .

• Computed by numerical root finding (Brent)

  w_L\ex = 4.018645...