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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