ex-1: reviewed

notes/sections/1.md

@@ -167,7 +167,7 @@ or *Robertson-Cryer* estimator was used. This estimator was chosen because makes
 no assumptions on the underlying distribution and is not computationally
 expensive. The HSM is obtained by iteratively identifying the half modal
 interval, which is the smallest interval containing half of the observation.
-Once the sample is reduced to less that three points the mode is computed as the
+Once the sample is reduced to less than three points the mode is computed as the
 average. The special case $n=3$ is dealt with by averaging the two closer points
 [@robertson74].
 
@@ -191,8 +191,8 @@ where $\sigma_e$ and $\sigma_o$ are the absolute errors of $m_e$ and $m_o$
 respectively. At 95% confidence level, the values are compatible if $p > 0.05$.
 In this case:
 
-  - t = 1.012
-  - p = 0.311
+  - $t = 1.012$
+  - $p = 0.311$
 
 Thus, the observed mode is compatible with the mode of the Landau distribution,
 although the result is quite imprecise.
@@ -246,7 +246,7 @@ $$
 
 for x, given a probability value $p_0$, where $p(x)$ is the CDF. The (unique)
 root of this equation was found by a root-finding routine
-(`gsl_root_fsolver_brent` in GSL) based on the Brent-Dekker method it too.
+(`gsl_root_fsolver_brent` in GSL) based on the Brent-Dekker method.
 The following condition was checked for convergence:
 $$
   |a - b| < \varepsilon_\text{abs} + \varepsilon_\text{rel} \min(|a|, |b|)
@@ -290,7 +290,7 @@ $$
 The function derivative $f'(x)$ was minimized using the same minimization method
 used for finding $m_e$. Once $f_\text{max}$ was known, the equation:
 $$
-  f'(x) = \frac{f_\text{max}}{2}
+  f(x) = \frac{f_\text{max}}{2}
 $$
 
 was solved by performing the Brent-Dekker method (described before) in the
@@ -303,8 +303,8 @@ $$
 \vspace{-1em}
 
 ![Example of a Moyal distribution density obtained by the KDE method. The rug
-  plot shows the original sample used in the reconstruction. The 0.6 factor
-  compensate for the otherwise peak height reduction.](images/1-landau-kde.pdf)
+  plot shows the original sample used in the reconstruction.
+](images/landau-kde.pdf)
 
 On the other hand, obtaining a good estimate of the FWHM from a sample is much
 more difficult. In principle, it could be measured by binning the data and