# B Bookdown cheat sheet

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

## B.1 Operation

R commands to do derivatives of a defined function Taking derivatives in R involves using the expression, D, and eval functions. You wrap the function you want to take the derivative of in expression(), then use D, then eval as follows.

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


a = b
(\#eq:test)

$$$a = b \tag{B.1}$$$

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{forx\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}

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