Math Equations

We do plenty of math, so I’d like to test out MathJax support.

Here is an example of MathJax inline rendering — $ 1/x^{2} $. And here is a block rendering:

\[r_{XY} = \frac{\mathrm{cov}(X,Y)}{\sqrt{\mathrm{var}(X)\mathrm{var}(Y)}}\]

Now, if we’d like to get serious, we’d do something involving multiline aligned equations, like

\[\begin{align} \mathcal{N}(t, \mu, \sigma) &= \mathrm{normal} \newline &= \frac{1}{\sqrt{2 \pi} \sigma} e^{-\frac{(t-\mu)^2}{2 \sigma^2}} \end{align}\]

or even an inline formula like $ \sum_{t=0}^{\infty} \frac{x^t}{t!} = e^x$.

Or we could try defining a command, like this. $ \newcommand{\water}{\mathrm{H}_{2}\mathrm{O}} $

Buffer slides off the sides of our tubes like \(\water\) off a duck’s back.

Or a more fancy set of equations:

\[\begin{align} \mbox{Union: } & A\cup B = \{x\mid x\in A \mbox{ or } x\in B\} \\ \mbox{Concatenation: } & A\circ B = \{xy\mid x\in A \mbox{ and } y\in B\} \\ \mbox{Star: } & A^\star = \{x_1x_2\ldots x_k \mid k\geq 0 \mbox{ and each } x_i\in A\} \\ \end{align}\]

Or to write the case likelihood function of PLCM model (Wu et al. 2015):

\[Pr(\boldsymbol{M}_i \mid I_i=1) = \sum_{\ell=1}^L\pi_\ell\theta_\ell^{M_{i\ell}}(1-\theta_\ell)^{1-M_{i\ell}}\prod_{j\neq \ell}\psi_j^{M_{ij}}(1-\psi_j)^{1-M_{ij}}\]

One can also use some doses of number theory…