######################
#MATH 4311 R Examples#
#                    #
#Transformations of  #
#Random Variables    #
######################



#Gamma Distribution
alpha=3
theta=0.01
X=rgamma(n=10000,shape=alpha,scale=theta)    
mean(X)
alpha*theta

sd(X)^2
alpha*theta^2


range(X)
index=seq(from=min(X),to=max(X),length.out=100)

hist(X,freq=FALSE,breaks=100)
lines(index,1/(gamma(alpha)*theta^alpha)*index^(alpha-1)*exp(-index/theta),col="red")


#Transforming X
Y=exp(X)


y.index=seq(from=min(Y),to=max(Y),length.out=100)

hist(Y,freq=FALSE,breaks=100)
lines(y.index,
      1/(gamma(alpha)*theta^alpha)*(log(y.index))^(alpha-1)/(y.index^(1+1/theta)),
      col="red")



#f(x)=3/32*x^5

X=(64*runif(10000))^(1/6)

x.index=seq(from=min(X),to=max(X),length.out=100)
hist(X,freq=FALSE,breaks=100)
lines(x.index,3/32*x.index^5,col="red")


#Tranformation
Y=X^3

y.index=seq(from=min(Y),to=max(Y),length.out=100)
hist(Y,freq=FALSE,breaks=100)
lines(y.index,1/32*y.index,col="red")

#########################
#How to write a function#
#########################
my.sum=function(a,b){
  c=a+b
  return(c)
}

my.sum(5,9)


hypotenuse=function(a,b){
  c=sqrt(a^2+b^2)
  return(c)
}

hypotenuse(3,4)


fibonacci=function(n){
  results=vector(length=n)
  results[1]=0
  results[2]=1
  for(i in 3:n){
    results[i]=results[i-1]+results[i-2]
  }
  return(results)
}

fibonacci(20)
fibonacci(1000)
f=fibonacci(1000)
f[1000]