PDF Reader

Adobe Reader

wget -c http://ardownload.adobe.com/pub/adobe/reader/unix/9.x/9.5.5/enu/AdbeRdr9.5.5-1_i386linux_enu.deb
sudo dpkg -i AdbeRdr9.5.5-1_i386linux_enu.deb

If you use 64-bit Ubuntu, you need install the following dependencies:

sudo apt-get install libxml2:i386 libstdc++6:i386

Now, you can run Adobe Reader by typing acroread in the terminal.

This post collects certain useful LeTeX commands, mainly focusing on user-defined ones.

Efficient Commands to Representing a Path in a Graph

% The etoolbox package is a toolbox of programming tools geared primarily towards
% LaTeX class and package authors. It provides LaTeX frontends to some of the new
% primitives provided by e-TeX as well as some generic tools which are not related to
% e-TeX but match the profile of this package.
\usepackage{etoolbox}
\ifx\walk\undefined 
\makeatletter
\newcommand{\walk}[1]{
  \@tempswafalse
  \@for\next:=#1\do
    {\if@tempswa\!\!\rightarrow\!\!\else\@tempswatrue\fi\next}
}
\makeatother
\fi

Notations

Rate Stable

Weakly Stable

Strongly Stable

For some Mac Air users, you might choose to use BasicTex instead of full-blown MaxTex for saving disk space or due to some other reasons. You might counter the following embarrassed situations, where your latex complained that some “sty” cannot be found. And when you trying to install it through tlmgr, tlmgr told you that it could not find the required file. At this time, you might need install the package (i.e., add the “sty” file) manually. This post will tell you how to add the “sty” by using “fullpage.sty” as an example.

Throughout the following of the post, we assume that you have already got the “sty” file.

If you are using old version Mac OS (e.g., Mac OS 10.7), please refer to this tutorial.

If you are using OS X EI Capitan or even newer, the following tutorial might be helpful.

First, Copy the “sty” file to the following directory:

/usr/local/texlive/2015basic/texmf-dist/tex/latex/base

For example,

sudo cp fullpage.sty /usr/local/texlive/2015basic/texmf-dist/tex/latex/base/

Then, run texhash to help latex to index the newly added “sty” file.

For example,

sudo texhash

Weak Law of Large Numbers

For any sequence $X_1, X_2, ...$ of i.i.d. random variables with finite expectation $\mathbb{E}[X_1] = \mu$ and finite variance $Var[X_1] = \sigma^2$, then the sample mean

converges to $\mu$ in probability, i.e.,

for any $\epsilon > 0$.

Proof: Using Chebyshev’s Inequality, the Weak Law of Large Numbers can be easily proved.

Strong Law of Large Numbers

For any sequence $X_1, X_2, ...$ of i.i.d. random variables with finite expectation $\mathbb{E}[X_1] = \mu$ and finite variance $Var[X_1] = \sigma^2$, then the sample mean

converges to $\mu$ almost surely, i.e.,