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50.2-bookdown_cheat_sheet.Rmd
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50.2-bookdown_cheat_sheet.Rmd
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# Bookdown cheat sheet
```{r}
# to see non-scientific notation a result
format(12e-17, scientific = FALSE)
```
## 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
```{r}
#define a function
f=expression(sqrt(x))
#take the first derivative
df.dx=D(f,'x')
df.dx
#take the second derivative
d2f.dx2=D(D(f,'x'),'x')
d2f.dx2
```
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.
```{r}
#evaluate the function at a given x
eval(f,list(x=3))
#evaluate the first derivative at a given x
eval(df.dx,list(x=3))
#evaluate the second derivative at a given x
eval(d2f.dx2,list(x=3))
```
## Math Expression/ Syntax
[Full list](http://tug.ctan.org/info/symbols/comprehensive/symbols-a4.pdf)
Aligning equations
```{r eval=FALSE}
\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
```{r, eval = FALSE}
\begin{equation}
a = b
(\#eq:test)
\end{equation}
```
```{=tex}
\begin{equation}
a = b
(\#eq:test)
\end{equation}
```
to refer in a sentence \@ref(eq:test) (`\@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
```{r eval = FALSE}
$$\begin{array}
{rrr}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}
$$
```
$$
\begin{array}
{rrr}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}
$$
```{r eval=FALSE}
$$\mathbf{X} = \left[\begin{array}
{rrr}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right]
$$
```
$$
\mathbf{X} = \left[\begin{array}
{rrr}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right]
$$
Aligning Equations
Aligning Equations with Comments
```{r eval=FALSE}
\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}
```
$$
\begin{aligned}
3+x &=4 & &\text{(Solve for} x \text{.)} \\
x &=4-3 && \text{(Subtract 3 from both sides.)} \\
x &=1 && \text{(Yielding the solution.)}
\end{aligned}
$$
### Statistics Notation
```{r eval = FALSE}
$$
f(y|N,p) = \frac{N!}{y!(N-y)!}\cdot p^y \cdot (1-p)^{N-y} = {{N}\choose{y}} \cdot p^y \cdot (1-p)^{N-y}
$$
```
$$
f(y|N,p) = \frac{N!}{y!(N-y)!}\cdot p^y \cdot (1-p)^{N-y} = {{N}\choose{y}} \cdot p^y \cdot (1-p)^{N-y}
$$
```{r eval=FALSE}
\begin{cases}
\frac{1}{b-a}&\text{for $x\in[a,b]$}\\
0&\text{otherwise}\\
\end{cases}
```
$$
\begin{cases}
\frac{1}{b-a} & \text{for } x\in[a,b]\\
0 & \text{otherwise}\\
\end{cases}
$$
## Table
```{r eval = FALSE}
+---------------+---------------+--------------------+
| 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** |
+--------------+--------------+----------------------+
```{r eval=FALSE}
(\mathbf{x}^T\mathbf{x})^{-1}\mathbf{x}^T\mathbf{y}
```
$(\mathbf{x}^T\mathbf{x})^{-1}\mathbf{x}^T\mathbf{y}$