Is there an absolutely continuous random variable with equal probability density and cumulative distribution functions in its support? Is it unique? What about in the discrete case?

  • Authors

    • Ernesto Veres-Ferrer Universitat de Valencia
    • JoseM Pavia Universitat de Valencia
    2014-06-02
    https://doi.org/10.14419/ijasp.v2i2.2563

  • Abstract

    This paper inquires about the existence and uniqueness of a univariate continuous random variable for which both cumulative distribution and density functions are equal and asks about the conditions under which a possible extrapolation of the solution to the discrete case is possible. The issue is presented and solved as a problem and allows to obtain a new family of probability distributions. The different approaches followed to reach the solution could also serve to warn about some properties of density and cumulative functions that usually go unnoticed, helping to deepen the understanding of some of the weapons of the mathematical statistician’s arsenal.

    Keywords: Cumulative Distribution Function; Density Function; Elasticity; Mathematical Statistics; Reversed Hazard Rate.

  • References

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  • Received date: 2014-04-27

    Accepted date: 2014-05-24

    Published date: 2014-06-02