# Cumulative distribution function

## 2018/03/31

suppressPackageStartupMessages(library(UsingR))
x <- father.son\$fheight
round(sample(x, 10), 1)
##  [1] 63.3 70.2 64.3 71.5 68.0 67.1 64.7 68.9 65.9 69.7

To define a distribution we compute, for all possible values of $$a$$, the proportion of numbers in our list that are below $$a$$. We use the following notation:

$F(a)\equiv Pr(x\leq a)$

This is called the cumulative distribution function (CDF). When the CDF is derived from data, as opposed to theoretically, we also call it the empirical CDF (ECDF). We can plot $$F(a)$$ versus a like this:

smallest <- floor( min(x) )
largest <- ceiling( max(x) )
values <- seq(smallest, largest,len=300)
heightecdf <- ecdf(x)
plot(values, heightecdf(values), type="l",
xlab="a (Height in inches)",ylab="Pr(x <= a)")