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Nondecreasing Lower Bound on the Poisson Cumulative Distribution Function for z Standard Deviations Above the Mean

  • M. Bondareva (a1)

Abstract

In this paper we discuss a nondecreasing lower bound for the Poisson cumulative distribution function (CDF) at z standard deviations above the mean λ, where z and λ are parameters. This is important because the normal distribution as an approximation for the Poisson CDF may overestimate or underestimate its value. A sharp nondecreasing lower bound in the form of a step function is constructed. As a corollary of the bound's properties, for a given percent α and parameter λ, the minimal z is obtained such that, for any Poisson random variable with the mean greater or equal to λ, its αth percentile is at most z standard deviations above its mean. For Poisson distributed control parameters, the corollary allows simple policies measuring performance in terms of standard deviations from a benchmark.

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Copyright

Corresponding author

Postal address: William E. Simon Graduate School of Business Administration, University of Rochester, Rochester, NY 14627, USA. Email address: mariya.bondareva@simon.rochester.edu

References

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[1] Adell, J. A. and Jodra, P. (2005). The median of the Poisson distribution. Metrika. 61, 337346.
[2] Kane, J. M. (2001). Monotonic approach to central limits. Proc. Amer. Math. Soc. 129, 21272133.
[3] Khattri, S. K. (2010). Three proofs of the inequality e < (1 + 1/n)n0.5. Amer. Math. Monthly 117, 273277.

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Nondecreasing Lower Bound on the Poisson Cumulative Distribution Function for z Standard Deviations Above the Mean

  • M. Bondareva (a1)

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