# This program illustrate the pdf and pmf can be interpreted
# as a limiting histogram
# Created by Kno, 2011.3.22

# Discrete
# Binomial  Y_1, ..., Y_k ~ Bin(n,p) for k=k1, k2, k3 respectively
k1<-10; k2<-100; k3<-1000; n<-10; p<-0.4 ;
y1<-rbinom(k1, n, p); y2<-rbinom(k2, n, p);y3<-rbinom(k3, n, p);
x<-2:n;
par(mfrow=c(2,2));
hist(y1); hist(y2); hist(y3);
plot(dbinom(x,n,p)~x);
summary(y3);

# Bernoulli  Bernoulli(p) = Bin(1,p)
k1<-10; k2<-100; k3<-1000; n<-1; p<-0.4 ;
y1<-rbinom(k1, n, p); y2<-rbinom(k2, n, p);y3<-rbinom(k3, n, p);
x<-0:n;
par(mfrow=c(2,2));
hist(y1); hist(y2); hist(y3);
plot(dbinom(x,n,p)~x);
summary(y3);

# When k -> \infinity, the distribution of Y does not change
# Central limiting theorem: Distribution of \bar{Y} or sum of Y's
# can be approximated by normal
k1<-10; k2<-100; k3<-1000; n<-1000; p<-0.4 ;
y1<-rbinom(k1, n, p); y2<-rbinom(k2, n, p);y3<-rbinom(k3, n, p);
x<-0:n;
par(mfrow=c(2,2));
hist(y1); hist(y2); hist(y3);
nx<-(-n):n;nx<-nx/20 +n*p;
plot(dnorm(nx,n*p,sqrt(n*p*(1-p)))~nx);
stem(y3);

# Continuous
# normal
k1<-10; k2<-100; k3<-1000;
y1<-rnorm(k1); y2<-rnorm(k2);y3<-rnorm(k3);

par(mfrow=c(2,2));
hist(y1); hist(y2); hist(y3);
nx<-(-100):100;nx<-nx/20
plot(dnorm(nx)~nx);
stem(y3);

par(mfrow=c(2,2));
qqnorm(y1); qqnorm(y2); qqnorm(y3);