R : Copyright 2005, The R Foundation for Statistical Computing
Version 2.1.1  (2005-06-20), ISBN 3-900051-07-0

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>  # 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) 
[1] 0.9750021
>  pnorm(1.96, lower.tail=F)              # 1-P(1.96)
[1] 0.02499790
>  pnorm(1.96, lower.tail=F, log.p=T)     # log(1-P(1.96))
[1] -3.688964
>  # just a simple check 
>  log(.05)
[1] -2.995732
>  # go far into tail
>  pnorm(100, lower.tail=F, log.p=T)      # -x*x/2 - log x - stuff                              
[1] -5005.524
>  # quantiles
>  qnorm(.025)
[1] -1.959964
>  qnorm(.95)
[1] 1.644854
>  # go far into tail
>  qnorm(1e-6)                            # one in a million
[1] -4.753424
>  qnorm(log(1e-6), log.p=T)
[1] -4.753424
>  #
>  # now Student's t
>  #
>  pt(1,df=1)                             # Cauchy symmetries
[1] 0.75
>  qt(.25,1)
[1] -1
>  df <- 5:10                             # can it do a vector?
>  pt(2.306,df)                           # vague memory of a critical value
[1] 0.9653740 0.9697000 0.9727465 0.9749998 0.9767299 0.9780977
>  qt(.975,df)
[1] 2.570582 2.446912 2.364624 2.306004 2.262157 2.228139
>  qt(1.e-4,10000)                        # far in tail, big df
[1] -3.720396
>  #
>  # gamma and Poisson
>  #
>  pgamma(2*log(2),1)                     # 1 - 2**(-2)
[1] 0.75
>  pgamma(4,3)                            # incomplete gamma to 4, shape 3
[1] 0.7618967
>  x <- 4
>  (1 + x + x*x/2)*exp(-x)                # check
[1] 0.2381033
>  2*qgamma(.95,df/2)                     # chi-square with manydf
[1] 11.07050 12.59159 14.06714 15.50731 16.91898 18.30704
>  lam = .3                               # Poisson rate
>  (1+lam)*exp(-lam)                      # Pr(X=0) + Pr(X=1)
[1] 0.9630637
>  pgamma(.3,2,lower.tail=F)              # upper tail
[1] 0.9630637
>  #
>  # beta and binomial
>  # 
>  pbinom(0:8,8,.4)                       # Binomial df, n=8, p=.4
[1] 0.01679616 0.10637568 0.31539456 0.59408640 0.82632960 0.95019264 0.99148032
[8] 0.99934464 1.00000000
>  p <- dbinom(0:8,8,.4)                  # Binomial probs, n=8, p=.4
>  p
[1] 0.01679616 0.08957952 0.20901888 0.27869184 0.23224320 0.12386304 0.04128768
[8] 0.00786432 0.00065536
>  cumsum(p)                              # check df
[1] 0.01679616 0.10637568 0.31539456 0.59408640 0.82632960 0.95019264 0.99148032
[8] 0.99934464 1.00000000
>  sum(p*log(p))                          # information
[1] -1.740283
>  pbinom(0:6,6,.4)                       # cumulatives
[1] 0.046656 0.233280 0.544320 0.820800 0.959040 0.995904 1.000000
>  pbinom(3,6,.4)                         # Pr(X=0)+Pr(X=1)+Pr(X=2)+Pr(X=3)
[1] 0.8208
>  pbeta(.4,4,3,lower.tail=F)             # using incomplete beta function
[1] 0.8208
>  #
>  # done
>  rm(list=ls())
>  q()