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

The parameters Î¸ 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 Î¼_{X}.

c. Assume r > 2. Find Ïƒ^{2}_{X}.

d. Show that if r â‰¤ 1, Î¼_{X} does not exist.

e. Show that if r â‰¤ 2, Ïƒ^{2}_{X} does not exist.

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