#######################################
#      Probability Distributions      #
#######################################

#Many probability distributions are available in Splus
#The online SPLUS manual lists them

#Go to the Help pulldown, select Online Manuals, then
#Guide to Statistics Volume 1.  This brings up a pdf
#file; select Chapter 3 "Probability".

#Select page 56, "Splus Probability Functions"
#to find out about the four types of probability
#function; they are

# The r-function r**** : used to produce a random sample
# The p-function p**** : used to produce the cdf
# The d-function d**** : used to produce the pmf/pdf
# The q-function q**** : used to produce the inverse cdf

#For the Normal or Gaussian distribution

#Random number generation

x.normalsample<-rnorm(1000,mean=0,sd=1)  #a sample of size 1000 from the N(0,1)
x.normalsample<-rnorm(5000,mean=10,sd=3)  #a sample of size 5000 from the N(10,3^2)

#To inspect this sample, use the "hist" histogram function

hist(x.normalsample)

#CDF

qvec<-seq(from=-2.5,to=2.5,by=0.1) #The values at which we want to work out the cdf.
pvec<-pnorm(qvec,mean=0,sd=1) #The cdf
plot(qvec,pvec,type="l")

#PDF

xvec<-seq(from=-2.5,to=2.5,by=0.1)
yvec<-dnorm(xvec,mean=0,sd=1) #The pdf
plot(xvec,yvec,type="l")

#Inverse cdf

pvec<-c(1:99)/100
qvec<-qnorm(pvec,mean=0,sd=1) #The inverse cdf
plot(pvec,qvec,type="l")

##############################################

#Illustrations of other distributions

#Student-t

xvec<-seq(from=-3.5,to=3.5,by=0.1)
yvec<-dt(xvec,df=5) #Five degrees of freedom
plot(xvec,yvec,type="l")

#To look at a comparison of different Student-t distributions

df.vec<-c(1,2,3,4,5,10,20,50)
plot(xvec,yvec,type="n")
for(i in 1:length(df.vec)){
	yvec<-dt(xvec,df=df.vec[i]) #Degrees of freedom determined by elements of df.vec
	lines(xvec,yvec)
}

#Chi-squared
xvec<-seq(from=0,to=20,by=0.1)
yvec<-dchisq(xvec,df=5) #Five degrees of freedom
plot(xvec,yvec,type="l")


#Fisher-F
xvec<-seq(from=0,to=10,by=0.1)
yvec<-df(xvec,df1=5,df2=8) #Five and ten degrees of freedom
plot(xvec,yvec,type="l")


#Discrete distributions are also available; here are the random
#number generators

#Binomial : rbinom(10,n=20,p=0.3)
#Poisson  : rpois(10,lambda=2)
#Geometric : rgeom(10,p=0.1) 

#Note the SPLUS Geometric distribution is slightly different from the standard one
#It records the number of "failures" BEFORE the first success in a Bernoulli
#sequence; to transform to the "standard" Geometric, add 1 to all elements
#in the sample