B Bookdown cheat sheet

# to see non-scientific notation a result
format(12e-17, scientific = FALSE)
#> [1] "0.00000000000000012"

B.1 Operation

R commands for taking derivatives of a defined function involve the expression, D, and eval functions. You wrap the function you want to differentiate in expression(), apply D() to take the derivative, and use eval() to compute the result.

simple example

#define a function
f=expression(sqrt(x))

#take the first derivative
df.dx=D(f,'x')
df.dx
#> 0.5 * x^-0.5

#take the second derivative
d2f.dx2=D(D(f,'x'),'x')
d2f.dx2
#> 0.5 * (-0.5 * x^-1.5)

Evaluate

• The first argument passed to eval is the expression you want to evaluate
• the second is a list containing the values of all quantities that are not defined elsewhere.

#evaluate the function at a given x
eval(f,list(x=3))
#> [1] 1.732051

#evaluate the first derivative at a given x
eval(df.dx,list(x=3))
#> [1] 0.2886751

#evaluate the second derivative at a given x
eval(d2f.dx2,list(x=3))
#> [1] -0.04811252

B.2 Math Expression/ Syntax

Full list

Aligning equations

\begin{aligned}
a & = b \\
X &\sim {Norm}(10, 3) \\
5 & \le 10
\end{aligned}

$$\begin{aligned} a & = b \\ X &\sim {Norm}(10, 3) \\ 5 & \le 10 \end{aligned}$$

Cross-reference equation

\begin{equation} 
a = b
(\#eq:test)
\end{equation}

$$\begin{equation} a = b \tag{B.1} \end{equation}$$

to refer in a sentence (B.1) (\@ref(eq:test))

Math Syntax	Notation
$\pm$	$\pm$
$\ge$	$\ge$
$\le$	$\le$
$\neq$	$\neq$
$\equiv$	$\equiv$
$^\circ$	$^\circ$
$\times$	$\times$
$\cdot$	$\cdot$
$\leq$	$\leq$
$\geq$	$\geq$
\propto	$\propto$
$\subset$	$\subset$
$\subseteq$	$\subseteq$
$\leftarrow$	$\leftarrow$
$\rightarrow$	$\rightarrow$
$\Leftarrow$	$\Leftarrow$
$\Rightarrow$	$\Rightarrow$
$\approx$	$\approx$
$\mathbb{R}$	$\mathbb{R}$
$\sum_{n=1}^{10} n^2$	$\sum_{n=1}^{10} n^2$
$$\sum_{n=1}^{10} n^2$$	$$\sum_{n=1}^{10} n^2$$
$x^{n}$	$x^{n}$
$x_{n}$	$x_{n}$
$\overline{x}$	$\overline{x}$
$\hat{x}$	$\hat{x}$
$\tilde{x}$	$\tilde{x}$
\check{}	$\check{}$
\underset{\gamma}{\operatorname{argmin}}	$\underset{\gamma}{\operatorname{argmin}}$
$\frac{a}{b}$	$\frac{a}{b}$
$\frac{a}{b}$	$\frac{a}{b}$
$\displaystyle \frac{a}{b}$	$\displaystyle \frac{a}{b}$
$\binom{n}{k}$	$\binom{n}{k}$
$x_{1} + x_{2} + \cdots + x_{n}$	$x_{1} + x_{2} + \cdots + x_{n}$
$x_{1}, x_{2}, \dots, x_{n}$	$x_{1}, x_{2}, \dots, x_{n}$
\mathbf{x} = \langle x_{1}, x_{2}, \dots, x_{n}\rangle$	$\mathbf{x} = \langle x_{1}, x_{2}, \dots, x_{n}\rangle$
$x \in A$	$x \in A$
$|A|$	$|A|$
$x \in A$	$x \in A$
$x \subset B$	$x \subset B$
$x \subseteq B$	$x \subseteq B$
$A \cup B$	$A \cup B$
$A \cap B$	$A \cap B$
$X \sim Binom(n, \pi)$	$X \sim Binom(n, \pi)$
$\mathrm{P}(X \le x) = \text{pbinom}(x, n, \pi)$	$\mathrm{P}(X \le x) = \text{pbinom}(x, n, \pi)$
$P(A \mid B)$	$P(A \mid B)$
$\mathrm{P}(A \mid B)$	$\mathrm{P}(A \mid B)$
$\{1, 2, 3\}$	$\{1, 2, 3\}$
$\sin(x)$	$\sin(x)$
$\log(x)$	$\log(x)$
$\int_{a}^{b}$	$\int_{a}^{b}$
$\left(\int_{a}^{b} f(x) \; dx\right)$	$\left(\int_{a}^{b} f(x) \; dx\right)$
$\left[\int_{\-infty}^{\infty} f(x) \; dx\right]$	$\left[\int_{-\infty}^{\infty} f(x) \; dx\right]$
$\left. F(x) \right|_{a}^{b}$	$\left. F(x) \right|_{a}^{b}$
$\sum_{x = a}^{b} f(x)$	$\sum_{x = a}^{b} f(x)$
$\prod_{x = a}^{b} f(x)$	$\prod_{x = a}^{b} f(x)$
$\lim_{x \to \infty} f(x)$	$\lim_{x \to \infty} f(x)$
$\displaystyle \lim_{x \to \infty} f(x)$	$\displaystyle \lim_{x \to \infty} f(x)$
Greek Letters
$\alpha A$	$\alpha A$
$\beta B$	$\beta B$
$\gamma \Gamma$	$\gamma \Gamma$
$\delta \Delta$	$\delta \Delta$
$\epsilon \varepsilon E$	$\epsilon \varepsilon E$
$\zeta Z \sigma $	$\zeta Z \sigma$
$\eta H$	$\eta H$
$\theta \vartheta \Theta$	$\theta \vartheta \Theta$
$\iota I$	$\iota I$
$\kappa K$	$\kappa K$
$\lambda \Lambda$	$\lambda \Lambda$
$\mu M$	$\mu M$
$\nu N$	$\nu N$
$\xi\Xi$	$\xi\Xi$
$o O$	$o O$
$\pi \Pi$	$\pi \Pi$
$\rho\varrho P$	$\rho\varrho P$
$\sigma \Sigma$	$\sigma \Sigma$
$\tau T$	$\tau T$
$\upsilon \Upsilon$	$\upsilon \Upsilon$
$\phi \varphi \Phi$	$\phi \varphi \Phi$
$\chi X$	$\chi X$
$\psi \Psi$	$\psi \Psi$
$\omega \Omega$	$\omega \Omega$
$\cdot$	$\cdot$
$\cdots$	$\cdots$
$\ddots$	$\ddots$
$\ldots$	$\ldots$

Limit P(\lim_{n\to \infty}\bar{X}_n =\mu) =1

$$P(\lim_{n\to \infty}\bar{X}_n =\mu) =1$$

Matrices

$$\begin{array}
{rrr}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}
$$

$$\begin{array} {rrr} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}$$

$$\mathbf{X} = \left[\begin{array}
{rrr}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right]
$$

$$\mathbf{X} = \left[\begin{array} {rrr} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right]$$

Aligning Equations

Aligning Equations with Comments

\begin{aligned}
    3+x &=4 && \text{(Solve for} x \text{.)}\\
    x &=4-3 && \text{(Subtract 3 from both sides.)}\\
    x &=1   && \text{(Yielding the solution.)}
\end{aligned}

$$\begin{aligned} 3+x &=4 & &\text{(Solve for} x \text{.)} \\ x &=4-3 && \text{(Subtract 3 from both sides.)} \\ x &=1 && \text{(Yielding the solution.)} \end{aligned}$$

B.2.1 Statistics Notation

$$
f(y|N,p) = \frac{N!}{y!(N-y)!}\cdot p^y \cdot (1-p)^{N-y} = {{N}\choose{y}} \cdot p^y \cdot (1-p)^{N-y}
$$

$$f(y|N,p) = \frac{N!}{y!(N-y)!}\cdot p^y \cdot (1-p)^{N-y} = {{N}\choose{y}} \cdot p^y \cdot (1-p)^{N-y}$$

\begin{cases}
\frac{1}{b-a}&\text{for $x\in[a,b]$}\\
0&\text{otherwise}\\
\end{cases}

$$\begin{cases} \frac{1}{b-a} & \text{for } x\in[a,b]\\ 0 & \text{otherwise}\\ \end{cases}$$

B.3 Table

+---------------+---------------+--------------------+
| Fruit         | Price         | Advantages         |
+===============+===============+====================+
| *Bananas*     | $1.34         | - built-in wrapper |
|               |               | - bright color     |
+---------------+---------------+--------------------+
| Oranges       | $2.10         | - cures scurvy     |
|               |               | - **tasty**        |
+---------------+---------------+--------------------+

Fruit	Price	Advantages
Bananas	$1.34	built-in wrapper bright color
Oranges	$2.10	cures scurvy tasty

(\mathbf{x}^T\mathbf{x})^{-1}\mathbf{x}^T\mathbf{y}

$(\mathbf{x}^T\mathbf{x})^{-1}\mathbf{x}^T\mathbf{y}$