Characterizing Distributions by Moment Existence and Tail Behavior

An actuary fits a Pareto with $\alpha =1.8$. Her manager asks for the variance. The correct answer: the variance does not exist. With $\alpha =1.8$, the mean exists ($\alpha >1$), but the second moment is infinite ($\alpha \le 2$). This is not an edge case, catastrophe and cyber losses genuinely have infinite higher moments.

HIGH-FREQUENCY: Pareto moment existence and the heavy/light classification by hazard rate are tested repeatedly.

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

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

The survival function decays like $x^{-\alpha }$, too slow to make $\int x^{k}f(x)dx$ converge when $k\ge \alpha$.

KEY: For Pareto, $\alpha$ acts as a ceiling on moment order. The $k$-th moment exists only when $k<\alpha$. "Does not exist" can be the correct exam answer.

Common mistakes

• Claiming lognormal is light-tailed because all moments exist. Moment existence and tail classification are different. The lognormal has all moments but no MGF. Correct test: MGF existence.
• 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.
• Confusing increasing hazard with heavy tail. IHR = light tail, not heavy. Weibull($\tau >1$) has IHR and is light-tailed.

Bottom line

• Pareto: $E[X^k]<\infty$ iff $k<\alpha$. Alpha is a ceiling on moment order (variance requires $\alpha >2$)
• Heavy-tailed: MGF does not exist for any $t>0$; survival decays slower than any exponential. Pareto, lognormal, Weibull($\tau <1$)
• Light-tailed: MGF exists for some $t>0$. Exponential, gamma, Weibull($\tau >1$)
• Lognormal is the key edge case: all moments finite, but still heavy-tailed (no MGF)

Exam shortcut

When asked to "characterize" a distribution, run four checks: (1) which moments exist? (2) does the MGF exist? (3) hazard rate direction? (4) mean excess increasing/decreasing? All four are related. For the lognormal (the favorite exam question) the answers are mixed: all moments, no MGF, eventually decreasing hazard. "No MGF = heavy tail." Fastest single test. "Pareto moments stop at alpha." "DHR = heavy, IHR = light, CHR = exponential."

The full lesson (about 1,632 words, 11 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 2d

Browse all free Exam FAM lessons or jump into free Exam FAM practice questions.