82 lines
1.3 KiB
Markdown
82 lines
1.3 KiB
Markdown
# MC simulations
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## In summary
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-----------------------------------------------------
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Landau Moyal
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----------------- ----------------- -----------------
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median $m_L\ex$ $m_M\ex (μ, σ)$
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mode $\mu_L\ex$ $\mu_M\ex (μ)$
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FWHM $w_L\ex$ $w_M\ex (σ)$
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-----------------------------------------------------
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## Moyal parameters
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A $M(x)$ similar to $L(x)$ can be found by imposing:
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\vspace{15pt}
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- equal mode
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$$
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\mu_M\ex = \mu_L\ex \approx −0.22278298...
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$$
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. . .
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- equal width
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$$
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w_M\ex = w_L\ex = \sigma \cdot a
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$$
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$$
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\implies \sigma_M \approx 1.1191486...
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$$
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## Moyal parameters
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:::: {.columns}
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::: {.column width=50%}
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:::
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::: {.column width=50%}
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:::
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::::
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## Moyal parameters
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This leads to more different medians:
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\begin{align*}
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m_M = 0.787... \thus &m_M = 0.658... \\
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&m_L = 1.355...
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\end{align*}
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## Compatibility test
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Comparing results:
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$$
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p = 1 - \text{erf} \left( \frac{t}{\sqrt{2}} \right)\ \with
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t = \frac{|x\ex - x\ob|}{\sqrt{\sigma\ex^2 + \sigma\ob^2}}
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$$
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- $x\ex$ and $x\ob$ are the expected and observed values
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- $\sigma\ex$ and $\sigma\ob$ are their absolute errors
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. . .
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At 95% confidence level, the values are compatible if:
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$$
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p > 0.05
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$$
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