Do All Continuous Probability Distributions Have Finite Variance?

Introduction

When analyzing the properties of continuous probability distributions, one fundamental concept is variance. Variance measures the spread or dispersion of a distribution. However, not all distributions possess a finite variance. This article explores the nuances of this property through a specific example, delving into the conditions under which a distribution has a finite variance and discussing its implications.

Continuous Probability Distributions and Moments

A continuous probability distribution is fully characterized by its probability density function (PDF), which describes the relative likelihood of different outcomes. A key aspect of a distribution is its moments, which provide information about the shape and behavior of the distribution.

First and Second Moments

The first moment of a distribution is its expected value (mean), and the second moment, when considered with the first moment, defines the variance. Variance is a second-order moment and can be expressed as:

[ text{Var}(X) E(X^2) - [E(X)]^2 ]

To ensure the variance is finite, both (E(X)) and (E(X^2)) must be finite.

A Specific Distribution Example

Consider the continuous probability distribution with the density function defined as:

[ f(x) begin{cases} frac{a}{x^{1 a}} x geq 1 1 text{otherwise} end{cases} ]

This function is non-negative and integrates to one, making it a valid density for any positive choice of the parameter (a).

Finite Mean Condition

For the mean (first moment) to be finite, the following integral must converge:

[ E(X) int_{-infty}^{infty} x f(x) , dx int_{1}^{infty} x cdot frac{a}{x^{1 a}} , dx ]

If we evaluate this integral, we find that it converges if and only if (a 1). Therefore, for (a in (0, 1]), the distribution is well-defined but has no finite expected value.

Finite Variance Condition

For the variance to be finite, the second moment must also be finite:

[ E(X^2) int_{-infty}^{infty} x^2 f(x) , dx int_{1}^{infty} x^2 cdot frac{a}{x^{1 a}} , dx ]

When evaluated, this integral converges if and only if (a 2). Thus, for (a in (1, 2]), the distribution has a finite mean but no finite variance.

Generalization

This result can be generalized. For any positive natural number (n), if (a in [n-1, n]), the distribution will have no (n^{text{th}}) moment but will have an (m^{text{th}}) moment for all (m n). This implies that the higher the moment, the further the parameter (a) can be from 1 before the corresponding moments diverge.

Conclusion

The example provided demonstrates that not all continuous probability distributions have finite variance. The conditions under which these moments exist are critical for understanding the behavior of such distributions. Understanding these properties helps in the analysis and application of different statistical models and methods.