Useful Latex Equations used in R Markdown for Statistics

stats_equations.Rmd

---
title: "Sample Equations used in Statistics"
output: html_document
---

###  Summations

### Without Indices

$\sum x_{i}$

$\sum x_{i}^2$

$\sum x_{i}y_{i}$

#### With Indices - Inline Form

$\sum_{i=1}^n x_{i}$

$\sum_{i=1}^n x_{i}^2$

$\sum_{i=1}^n x_{i}y_{i}$

#### With Indices - Display Form

$$\sum_{i=1}^n x_{i}y_{i}$$

### Independent Samples

$$\mu_{\bar{x_{1}} - \bar{x_{2}}} = \mu_{1} - \mu_{2}$$

$$\sigma_{\bar{x_{1}} - \bar{x_{2}}}^2 = \frac {\sigma_{1}^2}{n_{1}} + \frac{\sigma_{2}^2}{n_{2}}$$

$$\mu_{\hat{p}_{1} - \hat{p}_{2}} = p_{1} - p_{2}$$

$$\sigma_{\hat{p}_{1} - \hat{p}_{2}}^2 = \frac {p_{1}(1 - p_{1})}{n_{1}} + \frac {p_{2}(1 - p_{2})}{n_{2}}$$


### Pooled Sample Variance

$$s_{p}^2 = \frac {(n_{1} - 1)s_{1}^2 + (n_{2} - 1)s_{2}^2}{n_{1} + n_{2} - 2}$$

### Pooled Sample Proportion

$$\hat{p} = \frac {n_{1}\hat{p}_1 + n_{2}\hat{p}_{2}}{n_{1} + n_{2}}$$

### Chi-Square Test

$$\chi^2 = \sum \frac {(O - E)^2}{E}$$


### Correlations

$${SS}_{xx} = \sum (x - \bar{x})^2 = \sum x^2 - \frac {(\sum x)^2}{n}$$

$${SS}_{xy} = \sum (x - \bar{x})(y - \bar{y}) = \sum xy - \frac {(\sum x)(\sum y)}{n}$$

$$r = \frac {{SS}_{xy}}{\sqrt {{SS}_{xx}{SS}_{yy}}}$$


### Regression

#### Population Regression Line

$$E(y) = \alpha + \beta{x}$$

$$var(y) = \sigma^2$$

#### Least Squares Line

$$\hat{y} = a + bx$$

where 

$$b = \frac {{SS}_{xy}}{{SS}_{xx}}$$

and 

$$\bar{y} = a + b\bar{x}$$


#### Residual Sum of Squares

$$SSResid = \sum (y - \hat{y})^2 = \sum y^2 - a\sum y - b \sum xy$$

#### Standard Errors

$$s_{e} = \sqrt \frac {SSResid}{n - 2}$$

$$s_{b} = \frac {s_{e}}{\sqrt {{SS}_{xx}}}$$

$$s_{a + bx} = s_{e} \sqrt {1 + \frac {1}{n} + \frac {(x - \bar{x})^2}{{SS}_{xx}}}$$

for prediction:

$$se(y - \hat{y}) = s_{e} \sqrt {1 + \frac {1}{n} + \frac {(x - \bar{x})^2}{{SS}_{xx}}}$$


### Variance

$$SSTr = \frac {T_{1}^2}{n_{1}} + \frac {T_{2}^2}{n_{2}} + ... + \frac {T_{k}^2}{n_{k}} - \frac {T^2}{n}$$

$$SSTo = x_{1}^2 + x_{2}^2 + ... + x_{k}^2 - \frac {T^2}{n}$$

$$SSE = SSTo - SSTr$$