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Just a short post, to share some codes used to generate animated graphs, with R. Assume that we would like to illustrate the law of large number, and the convergence of the average value from binomial sample. We can generate samples  using&#62; n=200&#62; k=1000&#62; set.seed(1)&#62; X=matrix(sample(0:1,size=n*k,replace=TRUE),n,k)Each row  will be a trajectory of heads and tails. For each trajectory, define the mean , which will denote the mean of the first  values. Such a matrix can be computed using&#62; cummean=function(M){+ U=matrix(M[,1],nrow(M),1)+ for(i in 2:ncol(M)){+ U=cbind(U,(U[,i-1]*(i-1)+M[,i])/i)}+ return(U)+ }Define then&#62; Xbar=cummean(X)Now, to generate an animated gif, the way I usually do it is to generate graphs (png graphs) using a loop,&#62; S=trunc(10^seq(1,3,by=.05))&#62; for(j in 1:length(S)){+ s=S[j]+ Xhist=hist(Xbar[,s],breaks=seq(0,1,by=.05),plot=FALSE)+ nfile=paste("LLN-",100+j,".png",sep="")+ png(nfile,600,350)+ layout(matrix(c(3,0,1,2),2,2,byrow=TRUE), c(3,1), c(1,3), TRUE)+ plot(1:s,Xbar[1,1:s],type="l",col="light blue",ylim=0:1,xlab="",ylab="",axes=FALSE,+  xlim=c(10,k),log="x")+ axis(1)+ axis(2)+ for(i in 2:(n-1)) lines(1:s,Xbar[i,1:s],col="light blue")+ lines(1:s,Xbar[n,1:s],col="red",lwd=2)+ abline(v=s)+ barplot(Xhist$counts, axes=TRUE,horiz=TRUE,col="light green",xlim=c(0,n/2*1.05))+ dev.off()+ }I start at 100 because afterwards, when merging files, it is better to have (really) consecutive numbers, since sometimes, the lexical order is used, i.e. after 1 is 10, then 100, etc. Then I use Terminal commandsHere, the delay is in /100 seconds, and I use an infinite loop. The graph is hereIt is possible to use&#62; library(animation)&#62; ani.options(interval=.15)&#62; saveGIF({      })But the loop can be used also to generate several graphs, and to produce an animated graph in a pdf document (slides or lecture notes). The idea is to use the same code, but the output is here a pdf graph.&#62; S=trunc(10^seq(1,3,by=.1))&#62; for(j in 1:length(S)){+ s=S[j]+ Xhist=hist(Xbar[,s],breaks=seq(0,1,by=.05),plot=FALSE)+ nfile=paste("LLN-",j,".pdf",sep="")+ pdf(nfile,10,6)+ layout(matrix(c(3,0,1,2),2,2,byrow=TRUE), c(3,1), c(1,3), TRUE)+ plot(1:s,Xbar[1,1:s],type="l",col="light blue",ylim=0:1,xlab="",ylab="",axes=FALSE,+ xlim=c(10,k),log="x")+ axis(1)+ axis(2)+ for(i in 2:(n-1)) lines(1:s,Xbar[i,1:s],col="light blue")+ lines(1:s,Xbar[n,1:s],col="red",lwd=2)+ abline(v=s)+ barplot(Xhist$counts, axes=TRUE,horiz=TRUE,col="light green",xlim=c(0,n/2*1.05))+ dev.off()+ }We can then import them in LaTeX,\documentclass[a4]{article}\usepackage{graphicx}\usepackage{animate}\begin{document}\begin{center}\animategraphics[height=3.1in,palindrome]{1}{/Users/UQAM/LLN-}{1}{21}\end{center}\end{document}This will generate the following pdf file. This animate package is described in several forums, e.g. http://www.geogebra.org/...