Classes of Distributions and Their Relationships

An actuary fits an exponential, then a gamma, then a Weibull, then a Pareto, each time getting a better fit. These are not four unrelated models. The exponential is a special case of the gamma, which sits inside the generalized gamma, which also contains the Weibull. Understanding the hierarchy means you never memorize distributions in isolation.

HIGH-FREQUENCY: The exponential-gamma-Weibull hierarchy and distribution construction via transformations, mixing, and splicing are core testable topics.

If $X$ has CDF $F_X$, then $Y=X^{1/\tau }$ has:

$$F_Y(y)=F_X(y^{\tau })$$

Exponential to Weibull: Start with $X\sim \text{Exp}(\theta )$. Then $Y=X^{1/\tau }$ has survival function:

$$S_Y(y)=\exp \left[-{\left(\frac{y}{c}\right)}^{\tau }\right]$$

When $\tau =1$, Weibull reduces to exponential.

KEY: The exponential is the building block of the distribution hierarchy. Power-transform it to get Weibull, sum copies to get gamma, mix with a gamma rate to get Pareto.

Exponential to Gamma: Sum of $n$ i.i.d. Exp($\theta$) variables gives Gamma($n,\theta$). At $\alpha =1$, gamma is exponential.

Common mistakes

• Computing mixture variance as weighted average of component variances. The trap is 1,375,000 instead of 1,847,500. You must include the between-component term.
• Confusing spliced weights with probabilities. ${\alpha }_i$ is not $\mathrm{Pr}(\text{interval }i)$. The probability equals ${\alpha }_i$ times the integral of the component density over that interval.
• Forgetting Weibull at $\tau =1$ is exponential. This simplification saves time. The hazard becomes constant $1/\theta$.

Bottom line

• Exponential $\subset$ Gamma $\subset$ Generalized Gamma and Exponential $\subset$ Weibull $\subset$ Generalized Gamma, with lognormal a limiting case of generalized gamma.
• Weibull = power transform of exponential: $Y=X^{1/\tau }$ where $X\sim \text{Exp}$; Weibull at $\tau =1$ and gamma at $\alpha =1$ reduce to exponential.
• Power transform CDF rule: $F_Y(y)=F_X(y^{\tau })$, never $F_X(y^{1/\tau })$.
• Lognormal = exponentiated normal. Pareto = continuous mixture of exponentials with a gamma-distributed rate.

Exam shortcut

When you see a distribution relationship question, ask: is it a special case, a transformation, a mixture, or a splice? The exam rarely asks you to derive from scratch, it asks you to recognize and simplify. Knowing Weibull($\tau =1$) = exponential means you skip the gamma function. "Expo is the mother." Power-transform to Weibull. Sum to gamma. Mix to Pareto. Exponentiate normal to lognormal. "Mixture variance = within + between."

The full lesson (about 2,022 words, 13 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 2c

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