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: Recognize the exponential-gamma-Weibull relationships and compute finite-mixture moments.

For a positive random variable $X$ and $\tau > 0$, the power relationship $Y = X^{1/\tau}$ gives:

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

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

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

The Weibull scale is $c=\theta^{1/\tau}$. 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.

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.
• Keeping the old scale after taking a power. Taking the square root of an exponential variable with scale 9 gives Weibull scale 3, not 9.
• 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 power relationship, or a mixture? 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 1,767 words, 12 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 2c

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