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Lectures_my/MC_2016/Lecture1/mchrzasz.log

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          Lectures_my/MC_2016/Lecture1/mchrzasz.tex
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
\begin{frame}\frametitle{Let's test the $\sqrt{N}$, $\rm \color{RubineRed}{Lecture1/Euler\_number}$  }
\begin{frame}\frametitle{Let's test the $\sqrt{N}$}
 
\only<1>
{
$\Rrightarrow$ In the last example we measured the Euler number using different number of pseudo-experiments.\\
 
\begin{exampleblock}{}
A sum of large number random variables has always a Gaussian distributions, independent of what kind of PDF was used to generate the sample. The only requirement is that the PDFs have a finite expected value.	
\end{exampleblock}
\ARROW E(1.3) Using ROOT draw the Gauss distribution:
\ARROW E(1.4) Using ROOT draw the Gauss distribution:
\begin{equation}
\rho(x;\mu, \sigma) = \dfrac{1}{\sigma\sqrt{2\pi} } e^{ \frac{-(x-\mu)^2}{2\sigma^2}} \nonumber
\end{equation}
and calculate (for given $\mu$ and $\sigma$ the:
\begin{align*}
E(x_i)=1/2   \Rightarrow E(R_n)=n/2 \\
V(x_i)=1/12    \Rightarrow V(R_n)=n/12
\end{align*}
\ARROW E(1.4) Calculate the above.\\
\ARROW E(1.5) Calculate the above.\\
\ARROW From above we get:
\begin{equation}
\dfrac{R_n-n/12}{\sqrt{n/12}} \xrightarrow{N\to \infty} N(0,1).\nonumber
\end{equation}
aka we get a Gaussian distribution.
\ARROW For $n=12$ $\Rightarrow~(R_{12}-6)$ ''practical'' Gauss generator.\\
\ARROW Attention: This kind of generators are not good in reproducing the tails of the Gauss!\\
\ARROW E(1.5) Construct the Gauss generator based on the math above in ROOT.
\ARROW E(1.6) Construct the Gauss generator based on the math above in ROOT.
 
 
\end{frame}
 
\end{footnotesize}
\end{frame}
 
 
\begin{frame}\frametitle{Buffon needle, $\rm \color{RubineRed}{Lecture1/Heads\_tails}$  }
\begin{frame}\frametitle{Buffon needle}
\begin{small}
 
 
$\Rrightarrow$ Lets make this toy experiment and calculate the $\pi$ number.\\
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}\frametitle{Let's test the $\sqrt{N}$, $\rm \color{RubineRed}{Lecture1/Euler\_number}$  }
+\begin{frame}\frametitle{Let's test the $\sqrt{N}$}
 \only<1>
 {
 $\Rrightarrow$ In the last example we measured the Euler number using different number of pseudo-experiments.\\
 \begin{exampleblock}{}
 A sum of large number random variables has always a Gaussian distributions, independent of what kind of PDF was used to generate the sample. The only requirement is that the PDFs have a finite expected value.
 \end{exampleblock}
+\ARROW E(1.3) Using ROOT draw the Gauss distribution:
+\ARROW E(1.4) Using ROOT draw the Gauss distribution:
 \begin{equation}
 \rho(x;\mu, \sigma) = \dfrac{1}{\sigma\sqrt{2\pi} } e^{ \frac{-(x-\mu)^2}{2\sigma^2}} \nonumber
 \end{equation}
 and calculate (for given $\mu$ and $\sigma$ the:
 \begin{align*}
 E(x_i)=1/2   \Rightarrow E(R_n)=n/2 \\
 V(x_i)=1/12    \Rightarrow V(R_n)=n/12
 \end{align*}
+\ARROW E(1.4) Calculate the above.\\
+\ARROW E(1.5) Calculate the above.\\
 \ARROW From above we get:
 \begin{equation}
 \dfrac{R_n-n/12}{\sqrt{n/12}} \xrightarrow{N\to \infty} N(0,1).\nonumber
 \end{equation}
 aka we get a Gaussian distribution.
 \ARROW For $n=12$ $\Rightarrow~(R_{12}-6)$ ''practical'' Gauss generator.\\
 \ARROW Attention: This kind of generators are not good in reproducing the tails of the Gauss!\\
+\ARROW E(1.5) Construct the Gauss generator based on the math above in ROOT.
+\ARROW E(1.6) Construct the Gauss generator based on the math above in ROOT.
 \end{frame}
 \end{footnotesize}
 \end{frame}
+\begin{frame}\frametitle{Buffon needle, $\rm \color{RubineRed}{Lecture1/Heads\_tails}$  }
+\begin{frame}\frametitle{Buffon needle}
 \begin{small}
 $\Rrightarrow$ Lets make this toy experiment and calculate the $\pi$ number.\\

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