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).