# pqdfns        Sept 2007
 # 
 # demo of probability functions
 # pdist gives cdf P(x)
 # qdist gives quantile q(p) or P(q(p))=p
 # ddist gives density
 #
 #
 # first some normals
 #
 pnorm(1.96)                            # P(1.96) 
 pnorm(1.96, lower.tail=F)              # 1-P(1.96)
 pnorm(1.96, lower.tail=F, log.p=T)     # log(1-P(1.96))
 # just a simple check 
 log(.05)
 # go far into tail
 pnorm(100, lower.tail=F, log.p=T)      # -x*x/2 - log x - stuff                              
 # quantiles
 qnorm(.025)
 qnorm(.95)
 # go far into tail
 qnorm(1e-6)                            # one in a million
 qnorm(log(1e-6), log.p=T)
 #
 # now Student's t
 #
 pt(1,df=1)                             # Cauchy symmetries
 qt(.25,1)
 df <- 5:10                             # can it do a vector?
 pt(2.306,df)                           # vague memory of a critical value
 qt(.975,df)
 qt(1.e-4,10000)                        # far in tail, big df
 #
 # gamma and Poisson
 #
 pgamma(2*log(2),1)                     # 1 - 2**(-2)
 pgamma(4,3)                            # incomplete gamma to 4, shape 3
 x <- 4
 (1 + x + x*x/2)*exp(-x)                # check
 2*qgamma(.95,df/2)                     # chi-square with manydf
 lam = .3                               # Poisson rate
 (1+lam)*exp(-lam)                      # Pr(X=0) + Pr(X=1)
 pgamma(.3,2,lower.tail=F)              # upper tail
 #
 # beta and binomial
 # 
 pbinom(0:8,8,.4)                       # Binomial df, n=8, p=.4
 p <- dbinom(0:8,8,.4)                  # Binomial probs, n=8, p=.4
 p
 cumsum(p)                              # check df
 sum(p*log(p))                          # information
 pbinom(0:6,6,.4)                       # cumulatives
 pbinom(3,6,.4)                         # Pr(X=0)+Pr(X=1)+Pr(X=2)+Pr(X=3)
 pbeta(.4,4,3,lower.tail=F)             # using incomplete beta function
 #
 # done
 rm(list=ls())
 q()