Describe how changes in the parameters affect the distributions.

Severity models live and die by their parameters. Knowing which knob does what (scale, shape, both) is the difference between a 3-minute solve and 8 minutes of algebra.

Scale parameters. A parameter $\theta$ is a scale parameter when $Y = cX$ follows the same family with $\theta$ replaced by $c\theta$ and every other parameter unchanged. Practical consequences:

• Mean scales by $c$, variance by $c^2$.
• Coefficient of variation $\sqrt{\text{Var}}/E$ is unchanged.
• Skewness, kurtosis, and any ratio of central moments are unchanged.

• Exponential$(\theta)$: $\theta$ is scale. $E[X] = \theta$, $\text{Var}(X) = \theta^2$.
• Gamma$(\alpha, \theta)$: $\theta$ scale, $\alpha$ shape. $E[X] = \alpha\theta$, $\text{Var}(X) = \alpha\theta^2$, $\text{CV} = 1/\sqrt{\alpha}$.
• Pareto$(\alpha, \theta)$: $\theta$ scale, $\alpha$ shape. $E[X] = \theta/(\alpha-1)$ for $\alpha > 1$.
• Weibull$(\tau, \theta)$: $\theta$ scale, $\tau$ shape. $E[X] = \theta\,\Gamma(1+1/\tau)$.

Common mistakes

• Treating $\mu$ in lognormal as the mean. The mean is $\exp(\mu + \sigma^{2}/2)$.
• Inflating the shape parameter. Only $\theta$ scales; $\alpha$ (or $\tau$) is fixed.
• Quoting $\theta/(\alpha-1)$ when $\alpha \le 1$. The Pareto mean is infinite there.

Bottom line

• Scale parameter $\theta$: multiplying $X$ by $c$ gives the same family with $c\theta$. The k-th moment scales by $c^k$. CV and skewness unchanged.
• Shape parameter $\alpha$ (or $\tau$) controls tail thickness, skewness, and hazard direction, independent of scale; for Pareto, gamma, and Weibull, higher shape thins the tail.
• Gamma CV is $1/\sqrt{\alpha}$: higher $\alpha$ lowers CV and makes the density more bell-like, collapsing to exponential at $\alpha = 1$.
• Weibull with $\tau < 1$ gives a decreasing hazard and heavy tail; $\tau = 1$ is exponential.

Exam shortcut

Under inflation, multiply $\theta$ by $1+r$ and re-apply the mean formula. Shape is frozen. For lognormal, go straight to $E[X^{k}] = \exp(k\mu + k^{2}\sigma^{2}/2)$. Set $k = 1$ for mean, $k = 2$ for second moment. For Pareto, chant "minus one for mean, minus two for variance": $E[X] = \theta/(\alpha-1)$, $\text{Var}(X) = \theta^{2}\alpha/[(\alpha-1)^{2}(\alpha-2)]$.

The full lesson (about 1,852 words, 12 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 1a

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