An alternative to the lognormal distribution for modeling highly skewed

An alternative to the lognormal distribution for modeling highly skewed populations is the Pareto distribution with parameters θ and r. The probability density function is

The parameters θ and r may be any positive numbers. Let X be a random variable with this distribution.

a. Find the cumulative distribution function of X.

b. Assume r > 1. Find μX.

c. Assume r > 2. Find σ2X.

d. Show that if r ≤ 1, μX does not exist.

e. Show that if r ≤ 2, σ2X does not exist.

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