diff --git a/Lectures_my/MC_2016/Lecture5/mchrzasz.log b/Lectures_my/MC_2016/Lecture5/mchrzasz.log
index b5057e2..73fff51 100644
--- a/Lectures_my/MC_2016/Lecture5/mchrzasz.log
+++ b/Lectures_my/MC_2016/Lecture5/mchrzasz.log
@@ -1,4 +1,4 @@
-This is XeTeX, Version 3.1415926-2.5-0.9999.3 (TeX Live 2013/Debian) (format=xelatex 2015.4.1)  29 MAR 2016 17:28
+This is XeTeX, Version 3.1415926-2.5-0.9999.3 (TeX Live 2013/Debian) (format=xelatex 2015.4.1)  5 APR 2016 14:48
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+Overfull \hbox (12.27284pt too wide) in paragraph at lines 453--453
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diff --git a/Lectures_my/MC_2016/Lecture5/mchrzasz.pdf b/Lectures_my/MC_2016/Lecture5/mchrzasz.pdf
index b8b1a80..4dfe060 100644
--- a/Lectures_my/MC_2016/Lecture5/mchrzasz.pdf
+++ b/Lectures_my/MC_2016/Lecture5/mchrzasz.pdf
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diff --git a/Lectures_my/MC_2016/Lecture5/mchrzasz.tex b/Lectures_my/MC_2016/Lecture5/mchrzasz.tex
index 2c8f715..fd572a5 100644
--- a/Lectures_my/MC_2016/Lecture5/mchrzasz.tex
+++ b/Lectures_my/MC_2016/Lecture5/mchrzasz.tex
@@ -306,10 +306,10 @@
  \item Divide the $[0,1)$ into $k$ subdivisions:\\ \begin{align*}
  0=a_0<a_1<a_2<a_3<...<a_k=1
  \end{align*}
- \item Let $a_{n_i}={X_1,X_2,..X_n}$ be an series of elements in the interval $[a_{i-1}, a_i)$ (with $n_i$ elements). The $p_i=P{a_{i-1}<X<a_i} =a_i-a_{i-1}$.
+ \item Let $a_{n_i}={X_1,X_2,..X_n}$ be an series of elements in the interval $[a_{i-1}, a_i)$ (with $n_i$ elements). The $p_i=P({a_{i-1}<X<a_i}) =a_i-a_{i-1}$.
  \item A random variable:
  \begin{align*}
- \chi^2_k=\sum_{i=1}^k \dfrac{(n_1-np_i)^2}{np_i},~~~~n=\sum_{i=1}^kn_i,
+ \chi^2_k=\sum_{i=1}^k \dfrac{(n_i-np_i)^2}{np_i},~~~~n=\sum_{i=1}^kn_i,
  \end{align*}
  had a $\chi^2$ distribution of $k-1$ degrees of freedom.
  \end{itemize}
@@ -373,7 +373,7 @@
  \end{footnotesize}
  \end{frame}
 
-\begin{frame}\frametitle{Kolomogorox-Smirnow}
+\begin{frame}\frametitle{Kolmogorov - Smirnov}
  \begin{footnotesize}
 \ARROW The K-S test is used to check if a Random variable has pdf of a distribution $F$. The test is based on the difference between the two distributions:
 \begin{align*}
@@ -394,7 +394,7 @@
  \end{footnotesize}
  \end{frame}
 
-\begin{frame}\frametitle{Kolomogorox-Smirnow in practice}
+\begin{frame}\frametitle{Kolmogorov - Smirnov in practice}
  \begin{footnotesize}
 
 \begin{exampleblock}{ Take note:}
@@ -402,15 +402,15 @@
 \end{exampleblock}
 \ARROW In practice one should sort the numbers: $X_1,..., X_n$ and calculate the following:
 \begin{align*}
-D_n^{+}=\max_{1 \leq i\leq n} \left(\dfrac{i}{n} - F(X_{i:n} \right),~~~~D_n^{-}=\max_{1\leq i\leq n} \left(F(X_{i:n})- \dfrac{i-1}{n}\right)\\
+D_n^{+}=\max_{1 \leq i\leq n} \left(\dfrac{i}{n} - F(X_{i:n}) \right),~~~~D_n^{-}=\max_{1\leq i\leq n} \left(F(X_{i:n})- \dfrac{i-1}{n}\right)\\
 D_n=\max\lbrace D_n^+, D_n^- \rbrace
 \end{align*}
 where $X_{i:n}$ is so-called position statistic: $X_{1:n}, X_{2:n},..., X_{i:n}$.\\
 \ARROW The statistic $D_n$ asymptotically (in practice $n \geqslant 80$ ) is approaching the $\lambda$-Kolomogorows cdf:
 \begin{align*}
-\lim_{n \to \infty} \mathcal{P} \lbrace\sqrt{n}D_n \leqslant t \rbrace =K(t)= \sum_{j=-\infty}^{\infty} (-1)^j e^{-2j^2 t^2},~t>0
+\lim_{n \to \infty} \mathcal{P} \lbrace\sqrt{n}D_n  \leqslant t \rbrace =K(t)= \sum_{j=-\infty}^{\infty} (-1)^j e^{-2j^2 t^2},~t>0
 \end{align*}
-for which the critical values $\lambda_{\alpha}(\mathcal{P}\lbrace\sqrt{n}D_n>\lambda_{\alpha}$ can be found in the mathematical tables.\\
+for which the critical values $\lambda_{\alpha}(\mathcal{P}\lbrace\sqrt{n}D_n\rbrace ) >\lambda_{\alpha}$ can be found in the mathematical tables.\\
 \ARROW Commonly the $\lambda_{0.1}=1.224$, $\lambda_{0.05}=1.358$, $\lambda_{0.01}=1.628$ are used.
 
 
@@ -427,7 +427,7 @@
 \ARROW the random variables form the new pdf:
 \begin{align*}
 g_m(y)=\begin{cases}
-\dfrac{1}{m-1} \left[ y^{m-1} - {m \choose 1}(y-1)^{m-1} +{m \choose 2}(y-2)^{m-1}-..   \right]~~&{\rm for}  0\leq y\leq m,\\
+\dfrac{1}{m-1} \left[ y^{m-1} - {m \choose 1}(y-1)^{m-1} +{m \choose 2}(y-2)^{m-1}-..   \right]~~&{\rm for~}  0\leq y\leq m,\\
 0~~&{\rm else}
 \end{cases}
 \end{align*}