####### R Code for the Normalized Spacings of an Exponential Random Sample #######
#
# If X[1], X[2], ... X[n] are iid Exponential(theta) RVs, and we
# define their "Normalized Spacings" as
# 
# Y[j] = (n-j+1)[X(j) - X(j-1)], for j = 1, 2,..., n
#
# then Y[1], Y[2], ... Y[n] are also iid Exponential(theta) RVs.
#
# This result is called the Renyi Representation of Exponentials. 
# Intuitively, this result is a consequence of the Memoryless Proprty 
# of the Exponential Distribution.
#
# n = sample size
# mean.expo = Exponential expected value


Spacings.fn <- function(n, mean.expo){

### Generate a random sample of Exponential RVs ###

X <- rexp(n, rate = 1/mean.expo)

### Define the Normalized Spacings of the Sample ###

### Define the Order Statistics of the Sample ###

index <- c(rep(NA, n))
Y <- c(rep(NA, n))

Z <- c(0, sort(X))

for(j in 1:n){

	Y[j] <- (n - j + 1)*(Z[j+1] - Z[j])
}

for(j in 1:(n+1)){

	index[j] <- j-1

}

normalized.spacings <- c(0, Y)

order.stats <- Z
print(cbind(index, order.stats, normalized.spacings))

print(summary(X))
print(summary(Y))

par(mfrow = c(2,2))

hist(X, freq = FALSE, main = "Histogram of Exponential Data")
hist(Y, freq = FALSE, main = "Histogram of\n Normalized Spacings")

qqplot(x = X, y = Y, main = "QQ Plot",
       xlab = "Quantiles of the Exponential Data", 
	ylab = "Quantiles of the Normalized Spacings")
abline(a = 0, b = 1, lty = 1)

}

### Example ###

Spacings.fn(250, 7.5)