The Pareto random variable with parameters α >0 and β >0 has probability density function:
fX(x) = βα^β/[x^(β+1)], whereα < x < [infinity].
(a) Verify that fX is a density function.
(b) Find P{X > 2α}.
(c) Find the mean and variance of X. What restriction do you have on β in computing the mean and variance? Hint: There’s a different restriction on each.
(d) Use the probability transform to simulate 1000 Pareto random variables with α = 1 and β = 4 and find their sample mean and variance. Compare this to values in part (c).

Using pert, adam munson was able to determine that the expected project completion time for the construction of a pleasure yacht is 21 months, and the project variance is 4. a) what is the probability that the project will be completed in 17 months? b) what is the probability that the project will be completed in 20 months? c) what is the probability that the project will be completed in 23 months? d) what is the probability that the project will be completed in 25 months? e) what is the due date that yields a 95% chance of completion?