models.Rmd

---
title: "continous distributions"
author: "WILLIAM"
output: word-document
---
# Gamma
$$f(x) = \frac {x^{\alpha-1} \lambda^{\alpha} e^{-\lambda x}}{\Gamma\alpha}$$

## mean
$$E(x) = \frac{\alpha}{\lambda}$$

## Variance
$$Var(x) = \frac{\alpha}{\lambda^2}$$



# Beta
$$f(x) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha - 1}(1-x)^{\beta -1}$$

## mean
$$E(X)=\frac{\alpha}{\alpha+\beta}$$

## Variance
$$Var(x)=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$$

# Uniform
$$f(x)=\frac{1}{b-a}$$

## mean 
$$E(x)=\frac{a+b}{2}$$
## Variance
$$Var(x)=\frac{(b-a)^2}{12}$$
# Exponential
$$f(x)=\lambda e^{-\lambda x}$$
## mean
$$E(x)= \frac{1}{\lambda}$$
## Variance
$$Var(x)= \frac{1}{\lambda^2}$$
# Weibull
$$f(x)=\left(\frac{\alpha}{\theta}\right)\left(\frac{x}{\theta}\right)^{\alpha-1} e^{-({\frac{x}{\theta})^\alpha}}$$
## mean
$$E(x)=~~\Theta \Gamma(1+\frac{1}{\alpha})$$
## Variance
$$Var(x)=~~\Theta^2 \left(\Gamma(1+\frac{2}{\alpha})-\Gamma(1+\frac{1}{\alpha})^2\right)$$
# Normal
$$f(x)= \frac{1}{\sqrt{2 \pi \sigma^2}}e^{-\frac{1}{2}} \left(\frac{x - \mu}{\sigma}\right)^2 $$
## mean 
$$E(x)= \mu$$
## Variance
$$Var(x)=\sigma^2$$