Create new distributions by multiplication by a constant, raising to a power, exponentiation, mixing and splicing.

Five transformations let you build the entire severity zoo from a handful of base distributions. Inflation factors, Weibull severity fits, and lognormal claim sizes all fall out of these five moves.

Multiplication by a constant. Stretching the x-axis by $c$ stretches the CDF horizontally and squashes the pdf vertically by the same factor so total mass stays 1. If $Y = cX$ with $c > 0$, then $F_Y(y) = F_X(y/c)$ and $f_Y(y) = f_X(y/c)/c$. When $\theta$ is a scale parameter, the family is preserved and $\theta \to c\theta$. All shape parameters lock. Moments scale by $c^{k}$, and CV and skewness do not move.

KEY: Inflation at rate $r$ is multiplication by $1+r$. Replace $\theta$ with $(1+r)\theta$ and re-evaluate the mean formula. Shape parameters are frozen.

Raising to a power. Define $Y = X^{\tau}$. For $\tau > 0$ this is the "transformed" version; for $\tau < 0$ it is the "inverse" version.

Common mistakes

• Treating $Y = X^{\tau}$ as scaling. It is not; CV changes and shape parameters move.
• Forgetting the Jacobian. Missing the $|1/\tau| \cdot y^{1/\tau - 1}$ factor gives a non-density.
• Variance of a mixture using weighted variance. The correct route is $E[X^{2}] - (E[X])^{2}$, not $\sum w_i \sigma_i^{2}$.

Bottom line

• Multiplication $Y = cX$: same family, only scale moves $\theta \to c\theta$ and $E[X^{k}] \to c^{k}E[X^{k}]$; shape parameters stay fixed. Inflation tool.
• Power $Y = X^{\tau}$: exponential to Weibull (τ > 0); $Y = 1/X$ gives the inverse family (Pareto to inverse Pareto).
• Exponentiation $Y = e^{X}$: normal to lognormal with the same parameters.
• Mixing: weight pdfs by a mixing distribution, $\sum w_i f_i(x)$; gamma-mixed exponential = Pareto. Moments mix by the weights, but variance does not.

Exam shortcut

For a power transform, write $Y = X^{\tau}$, invert to $X = Y^{1/\tau}$, and apply change-of-variables once. For a discrete mixture, compute each $E_i[X^{k}]$ from the standard tables, then dot-product with $w_i$. For a two-piece splice with weights $w_1, w_2$, the scaling on piece $i$ is $w_i / [F_i(c_i) - F_i(c_{i-1})]$. Apply once, integrate to check. For "solve for the mixing weight" problems, plug into $w = (\mu^{\ast} - \mu_{2})/(\mu_{1} - \mu_{2})$; avoid setting up the full pdf integral.

The full lesson (about 4,068 words, 27 min read) adds 5 worked examples, all 11 common mistakes, a self-check, free in the app.

Learning objectives

• 1b

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