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

# binomial PMF and CDF in the tay-sachs case study

n <- 5

p <- 0.25

cbind( 0:n, dbinom( 0:n, n, p ), pbinom( 0:n, n, p ) )

     [,1]         [,2]      [,3]
[1,]    0 0.2373046875 0.2373047
[2,]    1 0.3955078125 0.6328125
[3,]    2 0.2636718750 0.8964844
[4,]    3 0.0878906250 0.9843750
[5,]    4 0.0146484375 0.9990234
[6,]    5 0.0009765625 1.0000000

par( mfrow = c( 2, 1 ) )

plot( 0, 0, xlab = 'y', ylab = 'Probability', type = 'n',
  xlim = c( -1, 6 ), ylim = c( 0, 1 ),
  main = 'Binomial PMF with n = 5 and p = 0.25' )

segments( 0, 0, 0, dbinom( 0, n, p ), lwd = 2, col = 'red' )

segments( 1, 0, 1, dbinom( 1, n, p ), lwd = 2, col = 'red' )

segments( 2, 0, 2, dbinom( 2, n, p ), lwd = 2, col = 'red' )

segments( 3, 0, 3, dbinom( 3, n, p ), lwd = 2, col = 'red' )

segments( 4, 0, 4, dbinom( 4, n, p ), lwd = 2, col = 'red' )

segments( 5, 0, 5, dbinom( 5, n, p ), lwd = 2, col = 'red' )

plot( 0, 0, xlab = 'y', ylab = 'Probability', type = 'n',
  xlim = c( -1, 6 ), ylim = c( 0, 1 ),
  main = 'Binomial CDF with n = 5 and p = 0.25' )

segments( -1, 0, 0, 0, lwd = 2, col = 'blue' )

segments( 0, pbinom( 0, n, p ), 1, pbinom( 0, n, p ), lwd = 2, col = 'blue' )

segments( 1, pbinom( 1, n, p ), 2, pbinom( 1, n, p ), lwd = 2, col = 'blue' )

segments( 2, pbinom( 2, n, p ), 3, pbinom( 2, n, p ), lwd = 2, col = 'blue' )

segments( 3, pbinom( 3, n, p ), 4, pbinom( 3, n, p ), lwd = 2, col = 'blue' )

segments( 4, pbinom( 4, n, p ), 5, pbinom( 4, n, p ), lwd = 2, col = 'blue' )

segments( 5, pbinom( 5, n, p ), 6, pbinom( 5, n, p ), lwd = 2, col = 'blue' )

par( mfrow = c( 1, 1 ) )

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

# exponential PDF, CDF, and inverse CDF

help( dexp )

lambda <- 2

par( mfrow = c( 3, 1 ) )

y.grid <- seq( 0, 2.5, length = 500 )

plot( y.grid, dexp( y.grid, lambda ), type = 'l', 
  col = 'red', lwd = 2, xlab = 'y', ylab = 'Density',
  main = 'Exponential PDF with lambda = 2' )

plot( y.grid, pexp( y.grid, lambda ), type = 'l', 
  col = 'blue', lwd = 2, xlab = 'y', ylab = 'Probability (p)',
  main = 'Exponential CDF with lambda = 2' )

abline( 0, 1, lwd = 2 )

p.grid <- seq( 0, 1, length = 500 )

plot( p.grid, qexp( p.grid, lambda ), type = 'l', 
  col = 'chocolate1', lwd = 2, xlab = 'p', ylab = 'Quantiles (y)',
  main = 'Exponential Inverse CDF with lambda = 2' )

par( mfrow = c( 1, 1 ) )

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