---
title: "Normal Parameters"
author: "Russell Almond"
date: "January 24, 2019"
output: html_document
runtime: shiny
---

```{r setup, include=FALSE}
library(shiny)
knitr::opts_chunk$set(echo = TRUE)
```

The [exponential distribution](https://en.wikipedia.org/wiki/Exponential_distribution) is a distribution often used for
waiting times.  Suppose the expected time to the next arrival is
$\theta$.  Then the probability that person will come at exactly time
$x$ is $f(x|\theta) = \frac{1}{\theta}e^{-x/\theta}$.  The exponential
distribution has some interesting properties.  In particular, if you
have already waited for time period $z$, then the conditional
expectation is $z+\theta$.

Suppose instead of waiting for one event, we wait for $k$ events.
Then we get the [gamma
distribution](https://en.wikipedia.org/wiki/Gamma_distribution) with
shape parameter $k$ and scale parameter $\theta$.  Its probability
density function is:
$$ f(x|k,\theta) = \frac{1}{\Gamma(k)\theta^k}x^{k-1}e^{-x/\theta}$$

The expected value is $k\theta$ and the standard deviation is $k\theta^2$.

```{r eruptions, echo=FALSE}
inputPanel(
  sliderInput("shape", label = "Shape parameter",
              min=0, max=15, value=3, step=1),
  
  sliderInput("scale", label = "Scale parameter",
              min = 0.2, max = 25, value = 10, step = 0.1)
)

renderPlot({
  shape <- as.numeric(input$shape)
  scale <- as.numeric(input$scale)
  curve(dgamma(x,shape,scale=scale),
        xlim=c(0,100),ylim=c(0,.1),
        main=paste("Gamma distribution with shape",shape,
                   "and scale",scale),
        xlab="X",ylab="Density")

})
```

Be somewhat careful when using the gamma distribution in R.  The gamma
distribution is often parameterized using the rate parameter
$\beta=1/theta$.  If you are using the scale parameter, you need to
name it explicitly and not rely on the position.

If the shape parameter is 1, then the gamma distribution is just the
exponential distribution.  It is extremely positively skewed.  As the
shape parameter increases, the gamma distribution becomes more and
more symmetric, eventually converging to the normal distribution.

The [chi-squared
distribution](https://en.wikipedia.org/wiki/Chi-squared_distribution)
is also a special case of the gamma distribution, with parameters
$k=\nu/2$ and $\theta=\nu$ (where $\nu$ is the degrees of freedom).
Therefore, the gamma distribution is often used to model variances.