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## 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:

```pnorm(-1)

[1]  0.1586553```

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 ...

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