n <- 3

m <- 114

print( theta.hat <- n / m )

binomial.likelihood.1 <- function( n, m, theta ) {

  likelihood <- theta^n * ( 1 - theta )^( m - n ) 

  return( likelihood )

}

binomial.likelihood.1( 3, 114, 0.1 )

[1] 8.335248e-09

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

plot( theta.grid.1, binomial.likelihood.1( n, m, theta.grid.1 ),
  type = 'l', ylab = 'Likelihood', xlab = 'theta' )

theta.grid.2 <- seq( 0, 0.1, length = 500 )

plot( theta.grid.2, binomial.likelihood.1( n, m, theta.grid.2 ),
  type = 'l', ylab = 'Likelihood', xlab = 'theta' )

binomial.log.likelihood.1 <- function( n, m, theta ) {

  log.likelihood <- n * log( theta ) + 
    ( m - n ) * log( 1 - theta ) 

  return( log.likelihood )

}

par( mfrow = c( 2, 1 ) )

plot( theta.grid.2, binomial.likelihood.1( n, m, theta.grid.2 ),
  type = 'l', ylab = 'Likelihood', xlab = 'theta' )

plot( theta.grid.2, binomial.log.likelihood.1( n, m, theta.grid.2 ),
  type = 'l', ylab = 'Log Likelihood', xlab = 'theta' )

par( mfrow = c( 1, 1 ) )

plot( theta.grid.1, dbeta( theta.grid.1, 1, 1 ), type = 'l',
  ylab = 'Density', xlab = 'theta', lwd = 2, ylim = c( 0, 3 ) )

lines( theta.grid.1, dbeta( theta.grid.1, 0.5, 1 ), lty = 2,
  lwd = 2, col = 'red' )

par( mfrow = c( 1, 1 ) )

plot( theta.grid.1, dbeta( theta.grid.1, 1, 1 ), type = 'l',
  ylab = 'Density', xlab = 'theta', lwd = 2, ylim = c( 0, 3 ) )

lines( theta.grid.1, dbeta( theta.grid.1, 0.5, 1 ), lty = 2,
  lwd = 2, col = 'blue' )

lines( theta.grid.1, dbeta( theta.grid.1, 0.1, 1 ), lty = 3,
  lwd = 2, col = 'blue' )

lines( theta.grid.1, dbeta( theta.grid.1, 0.01, 1 ), lty = 4,
  lwd = 2, col = 'green' )