## Continuous Probability Distributions

R has a wide range of built-in probability distributions, for each of which four functions are available: the probability density function (which has a d prefix); the cumulative probability (p); the quantiles of the distribution (q); and random numbers generated from the distribution (r). Each letter can be prefixed to the R function names in Table 7.1 (e.g. dbeta).

**Table 7.1**. The probability distributions supported by R. The meanings of the parameters are explained in the text.

The **cumulative probability** function is a straightforward notion: it is an S-shaped curve showing, for any value of *x*, the probability of obtaining a sample value that is less than or equal to *x*. Here is what it looks like for the normal distribution:

curve(pnorm(x),-3,3)
arrows(-1,0,-1,pnorm(-1),col="red")
arrows(-1,pnorm(-1),-3,pnorm(-1),col="green")

The value of *x*(−1) leads up to the cumulative probability (red arrow) and the probability associated with obtaining a value of this size (−1) or smaller is on the *y* axis (green arrow). The value on the *y* axis is 0.158 655 3:

The **probability densityis** the slope of this curve (its ‘derivative’). You can see at once that the slope is never negative. The slope starts out very shallow up to about *x* = −2, increases up to a peak (at *x* = 0 in this example) then gets shallower, and becomes very small indeed ...