diff --git a/Lectures_my/MC_2016/Lecture1/mchrzasz.log b/Lectures_my/MC_2016/Lecture1/mchrzasz.log index cac3ff0..dd7011c 100644 --- a/Lectures_my/MC_2016/Lecture1/mchrzasz.log +++ b/Lectures_my/MC_2016/Lecture1/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) 23 FEB 2016 16:40 +This is XeTeX, Version 3.1415926-2.5-0.9999.3 (TeX Live 2013/Debian) (format=xelatex 2015.4.1) 23 FEB 2016 16:59 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -3092,11 +3092,8 @@ [17 ] -Overfull \vbox (0.32344pt too high) detected at line 604 - [] - File: images/BG_lower.png Graphic file (type QTm) - + Overfull \vbox (19.18185pt too high) has occurred while \output is active [] ................................................. diff --git a/Lectures_my/MC_2016/Lecture1/mchrzasz.pdf b/Lectures_my/MC_2016/Lecture1/mchrzasz.pdf index 0600d11..68ca850 100644 --- a/Lectures_my/MC_2016/Lecture1/mchrzasz.pdf +++ b/Lectures_my/MC_2016/Lecture1/mchrzasz.pdf Binary files differ diff --git a/Lectures_my/MC_2016/Lecture1/mchrzasz.tex b/Lectures_my/MC_2016/Lecture1/mchrzasz.tex index 65f55e1..2c7916b 100644 --- a/Lectures_my/MC_2016/Lecture1/mchrzasz.tex +++ b/Lectures_my/MC_2016/Lecture1/mchrzasz.tex @@ -529,7 +529,7 @@ =0~~x,y~uncorrelated\\ >0~~x,y~correlated\\ <0~~x,y~anticorrelated\\ - \end{dcases} + \end{dcases}\nonumber \end{equation} \ARROW If x,y are independent then $cov(x,y) =0$ and $V(x+y) =V(x)+V(y)$\\ \ARROW If variables are independent then they are uncorrelated. If they are uncorrelated then can still be dependent$^{\dagger}$. @@ -549,7 +549,7 @@ The law of large numbers (LLN): let's take $n$ numbers from $\mathcal{U}(a,b)$ and for each of them we calculate the $f(u_i)$. The LLN: \begin{equation} -\frac{1}{n}\sum_{i=1}^nf(u_i) \xrightarrow{N\to \infty} \dfrac{1}{b-a}\int_a^b f(u)du +\frac{1}{n}\sum_{i=1}^nf(u_i) \xrightarrow{N\to \infty} \dfrac{1}{b-a}\int_a^b f(u)du \nonumber \end{equation} \ARROW We say (in statistic terminology) that the left side is asymptotically equivalent to the value of the integration if $n \to \infty$.\\ \ARROW Assumptions: @@ -591,7 +591,7 @@ \end{exampleblock} \ARROW E(1.3) Using ROOT draw the Gauss distribution: \begin{equation} -\rho(x;\mu \sigma) = \dfrac{1}{\sqrt{2}\pi } e^{ \dfrac{-(x-\mu)^2}{2\sigma^2}} \nonumber +\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*} @@ -626,7 +626,7 @@ \ARROW E(1.4) 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). +\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.\\