The following 20 iid random variable outcomes arose from some unknown cumulative distribution \(F(t)\) :

(a) Define the EDF for the random variable.

(b) Define the range of the random variable together with the probability density function that is implied by the EDF.

(c) Use the probability density function you derived in

(b) to define the mean and the variance of the random variable.

(d) Now ignore that there are "repeated observations" for some of the numbers in the table above and assign an equal weight of \(1 / 20\) to each of the 20 numbers. Repeat the calculation of the mean and the variance of the random variable acting as if each of the 20 numbers above are assigned \(1 / 20\) probability. Are the calculated mean and variance the same values? Should they be? Explain.

2 1 4 2 3 13 2 2 13 1 2 4 2 1 1 2 21