# Distribution functions invariant under residual-lifetime and length-biased sampling

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## Summary

*F*is a distribution function (d.f.), arises when the limiting d.f. of the

*residual-lifetime*in a renewal process is a scaled version of the

*general-lifetime*d.f.

*F.*The equation

*total-lifetime*in a renewal process is a scaled version of the

*general-lifetime*d.f. G.

For 0<*q*<1 the class, **F**_{ q }, of all d.f.'s satisfying (*) has been recently characterized and shown to include infinitely many d.f.'s. By explicitly exhibiting all the extreme points of **F**_{ q }, we recharacterize **F**_{ q } as the convex hull of its extreme points and use this characterization to show that for *q* close to one the d.f. solution to (*) is “*nearly unique.”* For example, if *q*>0.8 then all the infinitely many d.f.'s in **F**_{ q } agree to more than 15 decimal places.

The class, **G**_{ q }, of all d.f. solutions to (**) is studied here, apparently for the first time, and shown to be in a one-to-one correspondence with **F**_{ q }; symbolically, 1−F_{ q }(x) is the Laplace transform of G_{ q }(*qx*). For 0<*q*<1, we characterize G_{q} as the convex hull of its extreme points and obtain results analogous to those for **F**_{ q }. For *q*>1 we give a simple argument to show that neither (**) nor (*) has a d.f. solution. We present a complete, self-contained, unified treatment of the two dual families, **G**_{ q } and **F**_{ q }, and discuss previously known results.

A further application of the theory to graphical comparisons of two samples (*Q-Q* plots) is described.

### Keywords

Distribution Function Hull Stochastic Process Probability Theory Convex Hull## Preview

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### References

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