Tail Probabilities and Heavy-Tailed Distributions
Understand tail behavior and heavy-tailed distributions for Exam P.
Light vs. Heavy Tails
A distribution is light-tailed if its tail probabilities decrease exponentially fast (e.g., normal, exponential, gamma). A distribution is heavy-tailed if its tail decreases more slowly, typically as a power function. The Pareto distribution with S(x) = (theta/(theta+x))^alpha is the classic heavy-tailed example. Lognormal distributions are also considered heavy-tailed.
In insurance, heavy-tailed distributions model catastrophic losses, large liability claims, and reinsurance layers. Understanding tail behavior is essential for risk management and pricing.
Mean Excess Function
The mean excess function (or mean residual life) is e(d) = E[X - d | X > d]. For the exponential distribution, e(d) = 1/lambda (constant, due to the memoryless property). For the Pareto, e(d) = (theta + d)/(alpha - 1), which increases with d, indicating a heavy tail. An increasing mean excess function is a hallmark of heavy-tailed distributions.
On Exam P, you may be asked to compute the mean excess function or to use the relationship E[(X-d)+] = e(d) * S(d) = e(d) * P(X > d).
Tail Probability Calculations
For Exam P, be comfortable computing P(X > x) for common distributions. For the exponential: P(X > x) = e^(-lambda*x). For the Pareto: P(X > x) = (theta/(theta+x))^alpha. For the lognormal: P(X > x) = 1 - Phi((ln(x) - mu)/sigma). These tail probabilities directly determine the probability that a loss exceeds a deductible or a policy limit, making them central to insurance calculations.