Math and Equations in Rmarkdown

Greek Letters

We often use Greek letters when describing a statistical model. These can be rendered by using a backslash \ followed by the spelling of the greek letter you want. You can capitalize the spelling to get the uppercase version. For example:

• $\alpha$ - is \(\alpha\)
• $\beta$ - is \(\beta\)
• $\delta$ - is \(\delta\)
• $\gamma$ - is \(\gamma\)
• $\sigma$ - is \(\sigma\)
• $\Sigma$ - is \(\Sigma\)

and so on.

Fractions

Writing fractions is also pretty straight forward. We just use $\frac{}{}$ Where whatever we put in between the first set of brackets ends up in the numerator while the stuff we put in the second bracket goes to the denominator.

For example

$$\frac{(a + b)^2}{b^2}$$

will render as:

\[\frac{(a + b)^2}{b^2}\]

You may often have fractions within fractions:

$$\frac{(a + b)^2}{\Big(\frac{1}{2}a + \frac{1}{2}b\Big)}$$

\[\frac{(a + b)^2}{\Big(\frac{1}{2}a + \frac{1}{2}b\Big)}\]

Note the use of \Big( and \Big) for the parentheses. This tells math mode to make these parentheses larger to fit the equation we are making.

Sub Scripts and Super Scripts

You may have noticed the use of $b^2$ to make a superscript 2. We can _ and ^ to create sub and superscripts. Let’s try this out

$$a^2 + a^4 + a^8 + a^16$$

\[a^2 + a^4 + a^8 + a^{16}\]

hmmm, the last $a^16$ didn’t render properly. This is between the sub and superscript notation only looks at the first value. If you wish to make more than one value appear in a sub or super script you can wrap the values in braces {}

$$a^2 + a^4 + a^8 + a^{16}$$

\[a^2 + a^4 + a^8 + a^{16}\]

Regular Text

Sometimes you want to write some regualar words that don’t have any styling applied to them in an equation. We can applish this with the use of the keyword. See the following example

$$\text{My Equation} = \alpha + \beta\text{my_variable}$$

\[\text{My Equation} = \alpha + \beta\times\text{my_variable}\]

Summations

Summations are very common, and also very easy making use of the \sum keyword and the sub/superscript notation we just learned.

$$\text{mean value} = \sum_{i=1}^{n}\frac{x_i}{n}$$

\[\text{mean value} = \sum_{i=1}^{n}\frac{x_i}{n}\]

Common Math Notation

Here are some commonly used math notations

• Less than: $x \lt y$ - is \(x \lt y\)
• Greater than: $x \gt y$ - is \(x \gt y\)
• Greater than or equal to $x \ge y$ - is \(x \ge y\)
• Less than or equal to $x \le y$ - is \(x \le y\)
• Not equal to$x \ne y$ - is \(x \ne y\)
• Approximately equal to $x \approx y$ - is \(x \approx y\)
• Is proportional to $x \propto y$ - is \(x \propto y\)
• Is distributed as (squiggly line) $y \sim N(0, 1)$ - is \(y \sim N(0, 1)\)
• A hat for an estimate $\hat{y}$ - is \(\hat{y}\)
• A bar for an estimate $\bar{y}$ - is \(\bar{y}\)
• N choose k ${n \choose k}$ - \({n \choose k}\)
• More general use binom: $\binom{n}{k}$ - \(\binom{n}{k}\)
  • for example $\binom{n+1}{x + y + z}$ - \(\binom{n+1}{x + y + z}\)

Exercise 1.

Write out the probability mass function for the binomial distribution shown below. Try doing on your own. If you get stuck, you can see the solution by looking at the source code for this file. You can make the three dots between 2 and n as $\dots$ which renders as: \(\ldots\)

\[P(x; p, n) = \binom{n}{x}(p)^x(1-p)^{(n-x)}, \text{ for } x = 0, 1,2, \ldots, n\]

Arrays and Matrices

You can create arrays of numbers the \begin{array} and \end{array} keywords. For start an array with \begin{array} and we end an array with \end{array}. Within an array we add new columns with & and we add new rows with \\. Beside the begin array keyword we tell Rmarkdown the positioning of each column (c for center, l for left and r for right. For example, the following array has 3 rows, and 3 columns. Each entry is centered ({ccc})

$$\begin{array}{ccc}
x_{11} & x_{12} & x_{13}\\
x_{21} & x_{22} & x_{23} \\
x_{31} & x_{32} & x_{33} \\
\end{array}$$

This will render as

$$

Instead of an array we can create a matrix with square brackets using \begin{bmatrix} as:

$$X = \begin{bmatrix}
1 & x_{1}\\
1 & x_{2}\\
1 & x_{3}
\end{bmatrix}$$

\[X = \begin{bmatrix} 1 & x_{1}\\ 1 & x_{2}\\ 1 & x_{3} \end{bmatrix}\]

We can create a matrix with parentheses with \begin{pmatrix}, as in

$$X = \begin{pmatrix}
1 & x_{1}\\
1 & x_{2}\\
1 & x_{3}
\end{pmatrix}$$

\[X = \begin{pmatrix} 1 & x_{1}\\ 1 & x_{2}\\ 1 & x_{3} \end{pmatrix}\]

We can create a system of equations with the \being{aligned} keyword. Here we put an & symbol in the spot we wish to align. For example

$$\begin{aligned}
X + y &= a\timesx + a^2\timesb \\
x &= a(x - a\times b) - y
\end{aligned}$$

\[\begin{aligned} X + y &= (a\times x) + (a^2\times b) \\ X &= a(x - (a\times b)) - y \end{aligned}\]

Exercise 2

Try your best to recreate this equation for a random intercept and random slop model. If you need help, you can look at the source code.

\[y_i \sim N(\alpha_{j[i]} + \beta_1{j[i]} x_{1,i} + \beta_2 x_{2,i}, \sigma_y^2)\]

\[\begin{pmatrix} \alpha_j \\ \beta_j \end{pmatrix} = N\Big( \begin{pmatrix} \mu_\alpha \\ \mu_\beta \end{pmatrix}, \begin{pmatrix} \sigma_{\alpha}^2 & \rho\sigma_{\alpha}\sigma_{\beta} \\ \rho\sigma_{\alpha}\sigma_{\beta} & \sigma_{\beta}^2 \end{pmatrix}\Big), \text{ for } j = 1, \ldots, J\]