The
Inspection Paradox refers to the fact that in a Renewal Process, the length of
the interarrival period which contains a fixed time is stochastically larger
than the length of a typical interarrival period. To provide a more complete
understanding of this phenomenon, conditioning arguments are used to obtain the
distributions and moments of the lengths of
the interarrival periods other than the one containing this fixed time for the
case of the time-homogeneous Poisson Process. Distributions of the
waiting times for events that occur both before and after this fixed time are
derived. This provides a fairly complete probabilistic analysis of the
Inspection Paradox.
References
[1]
Field, S. (1991) Why Your Friends Have More Friends Than You Do. American Journal of Sociology, 96, 1464-1477. https://doi.org/10.1086/229693
[2]
Morozov, E. and Potakhina, L. (2014) An Application of the Inspection Paradox in Stability Analysis of Optical Systems. 2014 6th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), St. Petersburg, 6-8 October 2014, 522-525.
https://doi.org/10.1109/ICUMT.2014.7002156
[3]
Zelen, M. and Feinleib, M. (1969) On the Theory of Screening for Chronic Diseases. Biometrika, 56, 601-614. https://doi.org/10.1093/biomet/56.3.601
[4]
Lann, A. and Falk, R. (2006) Tell Me the Method, I’ll Give You the Mean. The American Statistician, 60, 322-327. https://doi.org/10.1198/000313006X151460
[5]
Ross, S.M. (2003) The Inspection Paradox. Probability in the Engineering and Informational Sciences, 17, 47-51. https://doi.org/10.1017/S0269964803171033
[6]
Pal, A., Kostinski, S. and Reuveni, S. (2022) The Inspection Paradox in Stochastic Resetting. Journal of Physics A: Mathematical and Theoretical, 55, 1-25.
https://doi.org/10.1088/1751-8121/ac3cdf
[7]
Karlin, S. and Taylor, H.M. (1975) A First Course in Stochastic Processes. 2nd Edition, Academic Press, San Diego.
https://doi.org/10.1016/B978-0-08-057041-9.50005-2
[8]
Ross, S.M. (1993) An Introduction to Probability Models. 5th Edition, Academic Press, San Diego. https://doi.org/10.1016/B978-0-12-598455-3.50004-0
[9]
Hogg, R.V., McKean, J.W. and Craig, A.T. (2018) Introduction to Mathematical Statistics. 8th Edition, Pearson, Boston.
