Using R for Introductory Statistics : John Verzani

Describing populations 1455.7 Toss two coins. Let X be the number of heads and Y the number of tails. Are X andY independent?5.2 Families of distributionsIn statistics there are a number of distributions that come in families. Each family isdescribed by a function that has a number of parameters characterizing the distribution.For example, the uniform distribution is a continuous distribution on the interval [a,b]that assigns equal probability to equal-sized areas in the interval. The parameters are aand b, the endpoints of the intervals.5.2.1 The d, p, q, and r functionsR has four types of functions for getting information about a family of distributions.The “d” functions return the p.d.f. of the distribution, whereas the “p” functions return thec.d.f. of the distribution. The “q” functions return the quantiles, and the “r” functionsreturn random samples from a distribution.These functions are all used similarly. Each family has a name and some parameters.The function name is found by combining either d, p, q, or r with the name for the family.The parameter names vary from family to family but are consistent within a family.For example, the uniform distribution on [a,b] has two parameters. The family nameis unif. In R the parameters are named min= and max=.> dunif(x=1, min=0, max=3)[1] 0.3333> punif(q=2, min=0, max=3)[1] 0.6667> qunif(p=1/2, min=0, max=3)[1] 1.5> runif(n=1, min=0, max=3)[1] 1.260The above commands are for the uniform distribution on [0, 3]. They show that thedensity is 1/3 at x=1 (as it is for all 0≤x≤3); the area to the left of 2 is 2/3; the median or.5-quantile is 1.5; and a realization of a random variable is 1.260. This last command willvary each time it is run.It is useful to know that the arguments to these functions can be vectors, in which caseother arguments are recycled. For example, multiple quantiles can be found at once.These commands will find the quintiles:> ps = seq(0,1,by=.2) # vector> names(ps)=as.character(seq(0,100,by=20)) # give names> qunif(ps, min=0, max=1)0 20 40 60 80 1000.0 0.2 0.4 0.6 0.8 1.0