# Example normal distribution

x <- rnorm(mean = 12, n = 150, sd = 5)
mean(x)
var(x)
1/150 * sum((x - mean(x))^2)
1/149 * sum((x - mean(x))^2)


# compare different estimates

means <- numeric(1000)
sdsnm1 <- numeric(1000)
sdsn <- numeric(1000)

for(i in 1:1000)
{
  x <- rnorm(mean  = 12, n = 50, sd  =5)
  means[i] <- mean(x)
  sdsnm1[i] <- 1/(length(x) -1) * sum((x - mean(x))^2)
  sdsn[i] <- 1/(length(x)) * sum((x - mean(x))^2)
}

mean(means)
mean(sdsnm1)
mean(sdsn)

par(mfrow = c(1,2))
boxplot(means)
abline( h = 12, lwd = 2, lty = 2)
dev.off()
boxplot(cbind(sdsnm1, sdsn), ylim = c(10,40))
abline( h = 25, lwd = 2, lty = 2)


boxplot(cbind(sqrt(sdsnm1), sqrt(sdsn)))
abline( h = 5, lwd = 2, lty = 2)

#------------
# unif-distribution U(0, theta)
# estimation of max -> theta

t1 <- numeric(1000)
t2 <- numeric(1000)

for(i in 1:1000)
{
  x <- runif(min  = 0, max = 10, n = 100)
  t1[i] <- mean(x)
  t2[i] <- ((length(x)+ 1)/length(x) * max(x))/2
}

mean(t1)
mean(t2)

summary(t2)
summary(t1)
boxplot(cbind(t1, t2))

hist(x, breaks = 50)