Chapter 19: CDF Tricks: Probability Plots and Random Number Generation

Quantiles and percentiles of a distribution can be found using the cumulative distribution function (CDF), which was defined in Chapter 15. For example, to find the 10th percentile of a distribution with CDF F, we solve F(x) = .1 for x. This percentile will exist and be unique for a continuous distribution. For many families of distributions, the quantiles are easy to find using R. For example, qgamma(.25,8,2) will return the 25th percentile of a Gamma(8,2) density, and qnorm(. 95,10, 2) will return the 95th percentile of a normal distribution with mean 10 and standard deviation 2. Further, qnorm(l :n/(n+l), 10, 2) will return n quantiles from this normal distribution, as shown for n = 20 in the plots below. On the left, the areas under the normal curve, between consecutive quantiles, are all equal to 1 /{n + 1). On the right, there are η equally spaced vertical ticks in (0,1), and these are mapped to the quantiles through the CDF.