#Create function to calculate expected value

EXP <- function(n) { if(n%%1 != 0)   { stop("Error: Input is not a positive integer") }
                 else if(n < 1)  { stop("Error: Input is not a positive integer") }
                 else if(n == 1) { 0 }
                 else            { log(n) + sum(choose(n,2:n)*(-1)^(2:n)*log(2:n)) } };


#Plot expected value function

N      <- 20;
NNN    <- (1:N);
EEE    <- rep(0, N);  for(n in 1:N) { EEE[n] <- EXP(n) };
DATA   <- data.frame(n = NNN, Expectation = EEE);
SPLINE <- as.data.frame(spline(NNN, EEE));

library(ggplot2);
ggplot(data = DATA, aes(x = n, y = Expectation)) +
  geom_point(colour = "DarkBlue") +
  geom_line(data = SPLINE, aes(x = x, y = y), colour = "Blue") +
  scale_x_continuous(name = "Number of Data Points", labels = (1:N), breaks = (1:N)) +
  ggtitle("Expected value of log-ratio of maximum-to-minimum") +
  labs(subtitle = "(Data from an exponential distribution)") +
  xlab("Number of Data Points") + ylab("Expected Value");