Characterizing Distributions by Moment Existence and Tail Behavior

An actuary fits a Pareto with $\alpha = 1.8$. Her manager asks for the variance. The fitted model has a finite mean because $\alpha > 1$, but an infinite second moment because $\alpha \leq 2$. Check which moments exist before using a variance-based calculation.

The existence of moments helps characterize which calculations a distribution supports. Verify convergence before reporting a mean, variance, or higher moment.

Classify distributions by which moments exist. When comparing probabilities of large losses, evaluate the specified models at the same finite threshold.

The Pareto k-th moment exists only when $k < \alpha$:

• $\alpha \leq 1$: mean infinite
• $1 < \alpha \leq 2$: mean exists, variance infinite
• $2 < \alpha \leq 3$: variance exists, skewness infinite
• $\alpha > k$: all moments through order $k$ exist

The moment integral $E[X^k]=\int_0^\infty x^k f(x)\,dx$ must converge. For Pareto, it diverges when $k \geq \alpha$, including equality.

Common mistakes

• Assuming equal means imply equal exceedance probabilities. Evaluate each survival function at the stated threshold with its own parameters.
• Stating Pareto mean exists for all $\alpha > 0$. The mean requires $\alpha > 1$. With $\alpha = 0.8$, the formula gives a negative number, the moment does not exist.
• Inferring moment existence from hazard direction. Gamma with shape below one has a decreasing hazard and finite moments. Pareto has a decreasing hazard too, but its moments require the order to be below its shape.

Bottom line

• Pareto moments exist only at orders strictly below the shape parameter.
• A finite Pareto mean requires shape above one; finite variance requires shape above two.
• Exponential, gamma, Weibull, and lognormal have finite moments of every positive integer order.
• Compare survival probabilities at the same specified threshold.

Exam shortcut

Check which moment integrals converge before substituting into a formula. For a requested exceedance probability, substitute the given threshold into the survival function. For an average excess, first check that the mean exists. For Pareto, compare the moment order directly with the shape parameter. Equality already gives an infinite moment.

The full lesson (about 1,489 words, 10 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 2d

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