Question

Suppose that you have programmed your computer to do the following:

(i) Draw 50 X values from a uniform distribution between 10 and 20. (i.e. uniform on [10,20])

(ii) Count the number of X values greater than 18 and call this number G

(iii) Divide G by 50 and call this number H

(iv) Repeat steps i to iii to obtain 10,000 H values

(v) Calculate the average H value and the variance of H Values

Answer the following questions:

1. What number should the average H value be, approximately?

2. What does the variance of H values estimate, intuitively?

3. How would average and variance of H values change if the procedure were repeated 100,000 times, instead of 10,000?

4. How would average and variance of H values change if 100 X values were drawn in step i, instead of 50?

Answer 1

Required R software code is given as follows, But please notice that your result may change as the random sample generated using R changes each time.

R-code

#(i).To generate 50 X values fron U[10,20]
X=runif(50,10,20)
X

#(ii). To Count the number of X values greater than 18
G=X[which(X>18)]
G
length(G) # Values are greater than 18

#(iii) We Divide G by 50
H=G/50
H

#(iv) WERepeat steps i to iii to obtain 10,000 H values
H=c()
for(i in 1:10000)
{
X[i]=runif(50,10,20)
G[i]=X[which(X>18)]
H[i]=G/50
}
H
length(H)

#(v) To Calculate the average H value and the variance of H Values
Average_H=mean(H)
Average_H

Variance_H=var(H)
Variance_H

Answers from my random sample

1. The average H value be 0.3 approximately
2. The variance of H values estimate the variation in H values which is zero intuitively
3. There will be no change in average and variance of H values change if the procedure were repeated 100,000 times, instead of 10,000
4. The average of H values and variance will still remain inchanged