## Wednesday, March 7, 2012

### Getting LaTeX in Blogger or Blogspot

As a someone who uses $\LaTeX$ quite a bit, I was excited to discover how to integrate it into my blogposts on blogger.com.  Since the idea of taking .pngs and putting them into the blog as images one by one is not ideal, I sought out scripts to install on blogger.com.  This post led me to the link here that got it up and running.  There are a lot of links out there that say 'just install this MathJax script' and it will work, but the problem is most of those links are no longer valid.  The one I just posted worked for me. Below I've listed one of my favorite theorems!

Central Limit Theorem

Let $X_{1}$, $X_{2}$, $\ldots$ be a sequence of iid random variables whose mgfs exist in a neighborhood of 0 (that is, $M_{X_{i}}(t)$ exists for $|t| < h$, for some $h \in \mathbb{R}^{+}$).  Let $E[X_{1}] = \mu_{X_{1}}$ and Var$[X_{1}] = \sigma_{X_{1}}^{2} > 0$, which are both finite since the mgf exists.  Then
$\frac{ \sum_{k=1}^{n} X_{k} - n\mu_{X_{1}} }{ \sigma_{X_{1}}\sqrt{n}} = \frac{ \bar{X} - \mu_{X_{1}} }{ \sigma_{X_{1}}/\sqrt{n} } \longrightarrow Z \sim \mathcal{N}(0,1) \text{ (converges in distribution)}$