Example R programs and commands The mth largest of k independent uniforms is distributed like beta(m,k+1-m) # All lines preceded by the "#" character are my comments. # All other left-justified lines are my input. # All other indented lines are the R program output. ### Theorem: ### If Xi~U([0,1]) is a uniform random variable for i-1,...,k, ### and X1,...Xk are independent, then Z=min(X1,...,Xk) has the ### distribution Z~Beta(1,k), whose p.d.f. is k*(1-t)^(k-1). ### More generally, if Z is the mth smallest for k i.i.d. uniforms, ### then Z~Beta(m,k+1-m). Here Beta(a,b) has p.d.f. C*t^(a-1)*(1-t)^(b-1). ### Illustrate this for some small k by sampling, taking mins ### and comparing the histogram to the graph of dbeta(): N<-10000 # number of mins to simulate x<-vector("numeric",length=N) # array to hold mins of samples t<-seq(0,1,by=0.01) # abscissa for the comparison p.d.f. plot # First trial: min of 5 i.i.d. uniforms k<-5 # number of independent uniforms for(n in 1:N) { x[n]=min(runif(5))} # main loop; runif() samples U([0,1]) # Plot the histogram hist(x,prob=T) # normalize to unit area for comparison with p.d.f. lines(t,dbeta(t,1,k)) # plot the Beta(1,k) p.d.f. on the same graph # Second trial: min of 9 i.i.d. uniforms k<-9 for(n in 1:N) { x[n]=min(runif(k))} hist(x,prob=T) lines(t,dbeta(t,1,k)) # Third trial: 3rd smallest of 9 i.i.d uniforms k<-9 m<-3 for(n in 1:N) { x[n]=sort(runif(k))[m]} # sort increasing; mth smallest hist(x,prob=T) lines(t,dbeta(t,m,k+1-m))