Question

Show how to compute the variance of a Pareto(α, θ) random variable. What is the condition on α for the variance to be defined?

Answer 1

The variance of a Pareto(α, θ) random variable is calculated using the specific formula for variance when α > 2. If α ≤ 2, the variance is undefined because the moments required for the variance to exist are infinite.

Step-by-step explanation:

The computation of the variance of a Pareto distribution with parameters α (alpha) and θ (theta) involves using the formula for the variance of a random variable. For the Pareto distribution, the variance is defined only if α > 2 and can be calculated using the formula:

Variance (V) = (θ^2 α) / ((α - 1)^2 (α - 2))

It is important to note that if α ≤ 2, the variance does not exist because the second moment of the distribution is infinite. When α > 2, this condition ensures that the required moments are finite for the variance to be well-defined.