A continuous random variable X has a Pareto distribution if its probability density function has the form

where a and λ are parameters (positive real numbers). The Pareto distribution is often used to model the distribution of incomes in a society. Exercises 56 and 57 involve the Pareto distribution.

Verify that for all positive numbers a and λ, the region under the graph of the Pareto probability density function f (x) has area 1, as required of all probability density functions.

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f(x)= = αλα xa+1 0 if x = x otherwise