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 μ_X.

c. Assume r > 2. Find σ²_X.

d. Show that if r ‰¤ 1, μ_X does not exist.

e. Show that if r ‰¤ 2, σ²_X does not exist.

Transcribed Image Text:

f(x) = { xr+l