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.
B.2 Math Expression/ Syntax
Aligning equations
\[ \begin{aligned} a & = b \\ X &\sim {Norm}(10, 3) \\ 5 & \le 10 \end{aligned} \]
Cross-reference 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
\[ \begin{array} {rrr} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} \]
\[ \mathbf{X} = \left[\begin{array} {rrr} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right] \]
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
$$
f(y|N,p) = \frac{N!}{y!(N-y)!}\cdot p^y \cdot (1-p)^{N-y} = {{N}\choose{y}} \cdot p^y \cdot (1-p)^{N-y}
$$\[ f(y|N,p) = \frac{N!}{y!(N-y)!}\cdot p^y \cdot (1-p)^{N-y} = {{N}\choose{y}} \cdot p^y \cdot (1-p)^{N-y} \]
\[ \begin{cases} \frac{1}{b-a} & \text{for } x\in[a,b]\\ 0 & \text{otherwise}\\ \end{cases} \]
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 |
|
| Oranges | $2.10 |
|
\((\mathbf{x}^T\mathbf{x})^{-1}\mathbf{x}^T\mathbf{y}\)