#1. Law of Large Numbers. Toss a coin repeatedly. 
#Plot #Heads/n vs. n.
#Plot #Heads vs. n.
#
#X_t = sum_k=1^t epsilon k.
#Plot X_t/t vs. t
#Plot X_t vs. t.
#
#
par(mfrow=c(2,2))
n=1000
p=.52
xbinom=rbinom(n,1,p)
plot(cumsum(xbinom)/(1:n),type="l",ylim=c(0,1),
xlab="#Tosses",ylab="Proportion of Heads",main="Proportion of Heads
p=.52")
abline(h=p)
#
plot(cumsum(xbinom),type="l",xlab="#Tosses",ylab="Total #Heads",main="#Heads,
.52(#Tosses)")
lines((1:n)*p)
#
plot(cumsum(xbinom)-(1:n)*p,type="l",xlab="#Tosses",ylab="#Heads -.52(#Tosses)",main="#Heads -
.52(#Tosses)")