## simulate some random data from the Binomial
n.samples <- 100000
x <- rbinom(n.samples,15,0.5)


# compute the emprical pdf
my.epdf <- matrix(0,16,2)
my.epdf[,1] <- 0:15
my.epdf[ as.numeric(names(table(x)))+1,2] <- table(x)/n.samples
my.epdf

# the theoretical pdf
my.theoreticalpdf <- dbinom(0:15,15,0.5)


#error in estimate
 my.epdf[,2]- my.theoreticalpdf

#relative error
(my.epdf[,2]  - my.theoreticalpdf )/my.theoreticalpdf


# compute the empirical cdf
my.ecdf <- matrix(0,16,2)
my.ecdf[,1] <- 0:15
my.ecdf[ as.numeric(names(table(x)))+1,2] <- cumsum(table(x)/n.samples)
my.ecdf[(max(as.numeric(names(table(x))))+1):16,2] <- 1
my.ecdf

# the theoretical cdf
my.theoreticalcdf <- pbinom(0:15,15,0.5)

#error in estimate
 my.ecdf[,2]- my.theoreticalcdf

#relative error
(my.ecdf[,2]  - my.theoreticalcdf )/my.theoreticalcdf


#plot the empirical pdf
barplot(table(x)/n.samples  ,xlim=c(0,15))

#plot the empirical cdf
plot(ecdf(x),ylim=c(0,15))








## simulate data from a continuos distribution


y <- runif(1000)

#plot empirical pdf with a smoother
hist(y,prob=TRUE)
lines(density(y))


#plot the empirical cdf
plot(ecdf(y))