Define the following set of numbers as "Saby's Favorite Random Numbers" (SFRN): {0.025, 0.075, 0.125, 0.175, ..., 0.875, 0.925, 0.975}. There are twenty distinct real numbers in the aforementioned set. You will need to review the Inverse Transform Method to solve parts of the problem below.

(a) Suppose Xi ~ Exponential( = 0.2). Use SFRN to generate 20 random numbers for X. Produce a sample set of 20 numbers{X1, X2, .., X20} and compute their average and standard deviation.

(b) Repeat part (a) using Excel's rand() function instead of SFRN.

(c) Suppose Y is a discrete random variable with the following probability distribution: P(Y = 1) = 0.1 P(Y = 2) = 0.5 P(Y = 3) = 0.25 P(Y = 4) = 0.1 P(Y = 5) = 0.05

Repeat part (a) for Y

(d) Repeat part (b) for Y (i.e. using Excel's rand() function instead of SFRN)

(e) Apply Monte Carlo simulation and use the 20 numbers in SFRN to estimate the size of the area that f(x,y) = x^2 + y^3 < 2, for 0 x 2 and 0 y 2. Specifically, please apply the 1st, 3rd, 5th, .., numbers to generate x, and the 2nd, 4th, 6th, ..., numbers to generate y, respectively.

(f) For part (e), please compute the 90% confidence interval if we apply a normal distribution