diff --git a/Lectures_my/MC_2016/Lecture8/mchrzasz.log b/Lectures_my/MC_2016/Lecture8/mchrzasz.log
index cd59fa2..46eb0b2 100644
--- a/Lectures_my/MC_2016/Lecture8/mchrzasz.log
+++ b/Lectures_my/MC_2016/Lecture8/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)  21 APR 2016 18:03
+This is XeTeX, Version 3.1415926-2.5-0.9999.3 (TeX Live 2013/Debian) (format=xelatex 2015.4.1)  21 APR 2016 18:15
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diff --git a/Lectures_my/MC_2016/Lecture8/mchrzasz.tex b/Lectures_my/MC_2016/Lecture8/mchrzasz.tex
index b521618..73ce3ec 100644
--- a/Lectures_my/MC_2016/Lecture8/mchrzasz.tex
+++ b/Lectures_my/MC_2016/Lecture8/mchrzasz.tex
@@ -639,7 +639,7 @@
 
 
 \end{frame}
-\begin{frame}\frametitle{Neumann-Ulam dual method, prove}
+\begin{frame}\frametitle{Neumann-Ulam dual method, proof}
 \begin{footnotesize}
 \ARROW If $Y_i(\gamma)$ is the i-th component of the $\overrightarrow{Y}(\gamma)$ vector. One needs to show:
 \begin{align*}
@@ -654,12 +654,12 @@
 \end{align*}
 \ARROW The expected value:
 \begin{align*}
-E \lbrace Y_j (\gamma)=\sum_{ {\rm trajectories}} \frac{a_j}{q_{i_0}p(0 \vert i_k) } P(i_1,i_2,...,i_k,0),
+E \lbrace Y_j (\gamma) \rbrace=\sum_{ {\rm trajectories}} \frac{a_j}{q_{i_0}p(0 \vert i_k) } P(i_1,i_2,...,i_k,0),
 \end{align*}
 where $P(i_1,i_2,...,i_k,0)$ is the probability of this trajectory occurring.\\
 \ARROW But by our definition the probability:
 \begin{align*}
-P(i_0,i_1,...,i_{k-1},j,0)=q_{i_k}h_{i_1,i_0}...h_{k,i_{k-1}}p(0 \vert j)
+P(i_0,i_1,...,i_{k-1},j,0)=q_{0}h_{i_1,i_0}...h_{k,i_{k-1}}p(0 \vert j)
 \end{align*}
 \ARROW In the end we get:
 \begin{align*}
@@ -748,7 +748,7 @@
 
 
 
-\begin{frame}\frametitle{Generalization, prove}
+\begin{frame}\frametitle{Generalization, proof}
 \ARROW For a $X(\gamma)$ trajectory the expected value is:
 \begin{align*}
 E \lbrace X(\gamma_k) = \sum_{k=0}^{\infty}\sum_{\gamma_k} X(\gamma_k)P \lbrace X(\gamma_k) \rbrace 
@@ -772,7 +772,7 @@
 
 \begin{frame}\frametitle{Generalization, the algorithm}
 \ARROW We set the $P$ matrix in a arbitrary way.\\
-\ARROW If in the $t$ moment the point is in the $i_t$ state the with the probability $p_{i_t, i_{t+1}}$ he can go to $i_{t+1}$ state. \\
+\ARROW If in the $t$ moment the point is in the $i_t$ state, then with the probability $p_{i_t, i_{t+1}}$ he can go to $i_{t+1}$ state. \\
 \ARROW We stop the walk once we reach $0$.\\
 \ARROW For the given trajectory we assign the value: $X(\gamma_k)$\\
 \ARROW We repeat the procedure $N$ times and take the mean and RMS.\\
@@ -792,7 +792,7 @@
 \ARROW W.Wasow (1956) was smarter:
 \begin{itemize}
 
-\item For the trajectory: $\gamma(i_0,i_1,...,i_k,0)$ we look at begging trajectories:
+\item For the trajectory: $\gamma(i_0,i_1,...,i_k,0)$ we look trajectories begging with:
 \begin{align*}
 (i_0),~(i_0,i_1),~(i_0,i_1,...,i_k)
 \end{align*}
@@ -808,9 +808,9 @@
 \end{itemize}
 \ARROW For the trajectory we define:
 \begin{align*}
-X^{\ast}(\gamma)=\sum_{m=0}^k nu_{i_1,i_2}...\nu_{i_{m-1},i_m}a_{i_m}
+X^{\ast}(\gamma)=\sum_{m=0}^k \nu_{i_1,i_2}...\nu_{i_{m-1},i_m}a_{i_m}
 \end{align*}
-\ARROW One can prove that:
+\ARROW One can proof that:
 \begin{align*}
 E \lbrace X^{\ast}(\gamma) \vert i_0=i \rbrace =x_i^0
 \end{align*}