# 5.2 Creating new distributions

Many continuous distributions are listed in Appendix A where formulas are given for the pdf, cdf, and other quantities. In actuarial applications, we are mainly interested in distributions that have only positive support, that is, where *F*(0) = 0. Distributions with this property that are familiar to most students of statistics include the exponential, gamma, Pareto, and lognormal distributions.

For any distribution, it is possible to construct other, new, distributions by making a transformation or by using some other mathematical technique. Many of the distributions in Appendix A can be obtained by applying such methods to other distributions that are also listed in Appendix A. In this section we illustrate such methods through numerous examples. The examples help explain some of the relationships between the distributions. In particular, Section 5.3 examines “families” of distributions where the members of the family are all special cases of a “parent” distribution.

## 5.2.1 Multiplication by a constant

This transformation is equivalent to applying inflation uniformly across all loss levels and is known as a change of scale. For example, if this year’s losses are given by the random variable *X*, then uniform inflation of 5% indicates that next year’s losses can be modeled with me random variable *Y* = 1.05*X*.

**Theorem 5.1** *Let X be a continuous random variable with pdf f*_{X} (*x*) *and cdf F*_{X}(*x*). *Let Y* = θ*X with* θ > 0. *Then*

**Proof:**

**Corollary 5.2** *The parameter ...*

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