# 14 Continuous RVs

## 14.1 Uniform (Continuous)

Models the scenario where all values in the unit interval [0,1] are equally likely.

$X \sim \mbox{Uniform}(0,1)$

$\mathcal{R} = [0,1]$

$f(x) = 1 \mbox{ for } x \in \mathcal{R}$

$F(y) = y \mbox{ for } y \in \mathcal{R}$

${\operatorname{E}}[X] = 1/2, \ {\operatorname{Var}}(X) = 1/12$

## 14.3 Uniform (Continuous) in R

> str(dunif)
function (x, min = 0, max = 1, log = FALSE)  
> str(punif)
function (q, min = 0, max = 1, lower.tail = TRUE, log.p = FALSE)  
> str(qunif)
function (p, min = 0, max = 1, lower.tail = TRUE, log.p = FALSE)  
> str(runif)
function (n, min = 0, max = 1)  

## 14.4 Exponential

Models a time to failure and has a “memoryless property”.

$X \sim \mbox{Exponential}(\lambda)$

$\mathcal{R} = [0, \infty)$

$f(x; \lambda) = \lambda e^{-\lambda x} \mbox{ for } x \in \mathcal{R}$

$F(y; \lambda) = 1 - e^{-\lambda y} \mbox{ for } y \in \mathcal{R}$

${\operatorname{E}}[X] = \frac{1}{\lambda}, \ {\operatorname{Var}}(X) = \frac{1}{\lambda^2}$

## 14.6 Exponential in R

> str(dexp)
function (x, rate = 1, log = FALSE)  
> str(pexp)
function (q, rate = 1, lower.tail = TRUE, log.p = FALSE)  
> str(qexp)
function (p, rate = 1, lower.tail = TRUE, log.p = FALSE)  
> str(rexp)
function (n, rate = 1)  

## 14.7 Beta

Yields values in $$(0,1)$$, so often used to generate random probabilities, such as the $$p$$ in Bernoulli$$(p)$$.

$X \sim \mbox{Beta}(\alpha,\beta) \mbox{ where } \alpha, \beta > 0$

$\mathcal{R} = (0,1)$

$f(x; \alpha, \beta) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) \Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta - 1} \mbox{ for } x \in \mathcal{R}$

where $$\Gamma(z) = \int_{0}^{\infty} x^{z-1} e^{-x} dx$$.

${\operatorname{E}}[X] = \frac{\alpha}{\alpha + \beta}, \ {\operatorname{Var}}(X) = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}$

## 14.9 Beta in R

> str(dbeta) #shape1=alpha, shape2=beta
function (x, shape1, shape2, ncp = 0, log = FALSE)  
> str(pbeta)
function (q, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE)  
> str(qbeta)
function (p, shape1, shape2, ncp = 0, lower.tail = TRUE, log.p = FALSE)  
> str(rbeta)
function (n, shape1, shape2, ncp = 0)  

## 14.10 Normal

Due to the Central Limit Theorem (covered later), this “bell curve” distribution is often observed in properly normalized real data.

$X \sim \mbox{Normal}(\mu, \sigma^2)$

$\mathcal{R} = (-\infty, \infty)$

$f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}} \mbox{ for } x \in \mathcal{R}$

${\operatorname{E}}[X] = \mu, \ {\operatorname{Var}}(X) = \sigma^2$

## 14.12 Normal in R

> str(dnorm) #notice it requires the STANDARD DEVIATION, not the variance
function (x, mean = 0, sd = 1, log = FALSE)  
> str(pnorm)
function (q, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)  
> str(qnorm)
function (p, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)  
> str(rnorm)
function (n, mean = 0, sd = 1)