R Notebook

#10000 random sample from exponential distribution of rate 3
x=rexp(1000,3)
e=numeric(1000)
#Making 1/(n-i+1)
c=1/(1000-1:1000+1)
#Expectation of order statistics
for (i in 1:1000) e[i]=sum(c[1:i]/3)
#Plotting expectation of order statistics 
plot(sort(x),e,col="blue")
abline(a=0,b=1,col="red")

If we don’t know parameter scale, we can estimate.

#10000 random sample from exponential distribution of rate 3
x=rexp(1000,3)
e=numeric(1000)
#Making 1/(n-i+1)
c=1/(1000-1:1000+1)
#Expectation of order statistics
for (i in 1:1000) e[i]=sum(c[1:i]*mean(x))
#Plotting expectation of order statistics 
plot(sort(x),e,col="blue")
abline(a=0,b=1,col="red")

For continous variables,

par(mfrow=c(2,2))
plot(sort(x),e,col="blue")
#Xk vrs F-1(K/n+1) n+1=1001
plot(sort(x),qexp(1:1000/1001,3),col="blue")

plot(pexp(sort(x),3),1:1000/1001,col="blue")

#Random chisq of df 2
x1=rchisq(1000,2)
par(mfrow=c(2,2))
plot(qnorm(1:1000/1001,mean = mean(x1),sd(x1)),sort(x1),col="blue")
#Same plot qq
qqnorm(x1)

We can’t say its normal.

#random 1000 values from uniform distribution
x2=runif(1000)
plot(sort(x2),qnorm(1:1000/1001,mean(x2),sd(x2)),col="blue")

not normal.

#1000 Random chisq with 40 df
x3=rchisq(1000,40)
plot(sort(x3),qnorm(1:1000/1001,mean(x3),sd(x3)),col="blue")

normal because chisq with higher degree move towards normal.

#1000 random normal sample 
x4=rpois(1000,4)
plot(sort(x4),qnorm(1:1000/1001,mean(x4),sd(x4)),col="blue")

because possion is not contious distribution.

#10000 random sample from normal dist
x5=rnorm(1000)
plot(sort(x5),qnorm(1:1000/1001,mean(x5),sd(x5)),col="blue")