diff --git a/Lectures_my/EMPP/Lecture1/images/result_error.pdf b/Lectures_my/EMPP/Lecture1/images/result_error.pdf
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+++ b/Lectures_my/EMPP/Lecture1/images/result_error.pdf
Binary files differ
diff --git a/Lectures_my/EMPP/Lecture1/mchrzasz-blx.bib b/Lectures_my/EMPP/Lecture1/mchrzasz-blx.bib
new file mode 100644
index 0000000..9e405b3
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+++ b/Lectures_my/EMPP/Lecture1/mchrzasz-blx.bib
@@ -0,0 +1,11 @@
+@Comment{$ biblatex control file $}
+@Comment{$ biblatex version 2.3 $}
+Do not modify this file!
+
+This is an auxiliary file used by the 'biblatex' package.
+This file may safely be deleted. It will be recreated as
+required.
+
+@Control{biblatex-control,
+  options = {2.3:0:0:1:0:0:1:1:0:0:0:0:1:1:3:1:79:+},
+}
diff --git a/Lectures_my/EMPP/Lecture1/mchrzasz.tex b/Lectures_my/EMPP/Lecture1/mchrzasz.tex
index d3fc7eb..94d8005 100644
--- a/Lectures_my/EMPP/Lecture1/mchrzasz.tex
+++ b/Lectures_my/EMPP/Lecture1/mchrzasz.tex
@@ -266,7 +266,7 @@
 
 \begin{frame}\frametitle{Course Plan}
 
-We will have 6 hourse of Monte Calro (MC) lectures. The lecutres will be devoted:\\
+We will have 6 hours of Monte Carlo (MC) lectures. The lectures will be devoted:\\
 
 \hspace{1.5cm}
 \begin{itemize}
@@ -277,7 +277,10 @@
 \item 1.5h: Hands-on tutorial with MC methods.
 \end{itemize}
 \hspace{1cm} \\
-The hands-on tutorial will consist of program templates in which you will have to implement couple of algoriths that were explained in the lecture. The solutions will be discussed on the last lecture.
+The hands-on tutorial will consist of program templates in which you will have to implement couple of algorithms that were explained in the lecture. The solutions will be discussed on the last lecture.\\
+$\Rrightarrow$ All examples shown in this course are available in the github repository:\\
+\url{https://github.com/mchrzasz/EMPP_MC}\\
+\color{RubineRed}{There will be an indication (in this color) on the adequate slide for each of the macro.}
 
 
 \end{frame} 
@@ -331,7 +334,7 @@
 
 
 
-\begin{frame}\frametitle{Euler number determination}
+\begin{frame}\frametitle{Euler number determination, $\rm \color{RubineRed}{Lecture1/Euler\_number}$  }
 \begin{footnotesize}
 $\Rrightarrow$ As mentioned before \mc~methods can be used to solve problems that \textbf{do not} have stochastic nature! All the integrals calculated in Los Alamos during the Manhattan project are nowadays solvable without any \mc~methods.\\
 $\rightarrowtail$ Let's give a trivial example of solving a non stochastic problem: calculating Euler number $e$. We know that $e=2.7182818...$.
@@ -347,8 +350,8 @@
 \end{align*}
 \end{itemize}
 $\Rrightarrow$ Numerical example:
-\begin{tabular}{r c c}
-$N$ & $\hat{e}$ & $\hat{e} - e$\\
+\begin{tabular}{r c c c }
+$N$ & $\hat{e}$ & $\hat{e} - e$ & \multirow{5}{*}{Is this $\sim\sqrt{N}$?} \\
 100 & $2.760000$ & $0.041718$ \\
 10000 & $2.725000$ & $0.006718$ \\
 1000000  & $2.718891$ & $0.000609$ \\
@@ -359,6 +362,36 @@
 \end{footnotesize}
 \end{frame}
 
+\begin{frame}\frametitle{Let's test the $\sqrt{N}$, $\rm \color{RubineRed}{Lecture1/Euler\_number}$  }
+
+\only<1>
+{
+$\Rrightarrow$ In the last example we measured the Euler number using different number of pseudo-experiments.\\
+$\rightarrowtail$ We compared the obtained value to the true and observed roughly a $\sqrt{N}$ dependence on the difference between the true value and the obtained one.\\
+}
+$\rightarrowtail$ Could we test this? YES! Lets put our experimentalist hat on!\\
+$\rightarrowtail$ From the begging of studies they tooth us to get the error you need to repeat the measurements, so let's do that:
+
+\begin{center}
+\only<2>
+{
+\includegraphics[angle=-90,width=0.8\textwidth]{images/result_error.pdf}
+}
+\end{center}
+
+
+
+
+
+
+
+
+
+
+
+
+\end{frame}
+
 \begin{frame}\frametitle{Monte Carlo and integration}
 \begin{footnotesize}
 $\hookrightarrow$ {\color{BrickRed}{\textbf{All MC calculations are equivalent to preforming an integration.}}}\\
@@ -521,8 +554,13 @@
 \end{footnotesize}
 \end{frame}
 
-\begin{frame}\frametitle{Large Number Theorem}
+\begin{frame}\frametitle{Central Limit Theorem}
 \begin{footnotesize}
+    \begin{exampleblock}{}                                                                                                                                                                                                                                                                                                
+Large independent random numbers assembly has always Gaussian distribution no matter from what distribution they were generated from as far as they have finite variances and expected values and the assembly is sufficiently large.
+        \end{exampleblock} 
+\includegraphics[width=0.9\textwidth]{images/dupa.png}
+
 
 \end{footnotesize}
 \end{frame}
@@ -530,7 +568,7 @@
 
 \begin{frame}\frametitle{Crude Monte Carlo method of integration}
 \begin{footnotesize}
-$\Rrightarrow$ {\color{MidnightBlue}{Crude Monte Carlo method of integration is based on Large Number Theorem (LNT): }}\\
+$\Rrightarrow$ {\color{MidnightBlue}{Crude Monte Carlo method of integration is based on Central Limit Theorem (CLT): }}\\
 \begin{align*}
 \dfrac{1}{N} \sum_{i=1}^N f(x_i) \xrightarrow{N\to \infty} \dfrac{1}{b-a}\int_a^b f(x)dx =E(f)
 \end{align*}
@@ -780,6 +818,30 @@
 \end{frame}
 
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}\frametitle{Wrap up}
+\begin{footnotesize}
+$\Rrightarrow$ To sum up:
+\begin{itemize}
+\item We discussed basic mathematical properties of \mc~methods.
+\item We shown that besides the stochastic nature of \mc~ they can be used to determine totally non stochastic quantities.
+\item We demonstrated there is a perfect isomorphism between \mc~method and integration.
+\item We learned how co calculate integrals and estimate the uncertainties.
+\item Finally we discussed several classical methods of variance reduction.
+\end{itemize}
+
+
+\end{footnotesize}
+\end{frame}
+
+
+
+
+
+
+
+
 \backupbegin   
 
 \begin{frame}\frametitle{Backup}