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 1,465 words, 10 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 1a

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