Hile the Pareto distributions are continuous, they tend to be used to model discrete data in humanities and actuarial sciences. Moreover, with its roots in power functions, Pareto distributions may be used in the growing popularity of the studies of networks. The probability density function (PDF) for a Pareto distribution is f(x) = α /x^(α+1), 1 ≤ x
where α > 0 is the shape parameter.
(a) Derive the cumulative distribution function (CDF)F(x) = 1−1/(x^α), 1 ≤ x.
(b) Derive the expected value μ = α /(α−1) for α > 1.
(c) Derive the standard deviation σ = √[α / ((α−1)^2(α−2))] for α > 2.