# Improvements of Polya Upper Bound for Cumulative Standard Normal Distribution and Related Functions

@inproceedings{Eidous2022ImprovementsOP, title={Improvements of Polya Upper Bound for Cumulative Standard Normal Distribution and Related Functions}, author={Omar M. Eidous}, year={2022} }

Although there is an extensive literature on the upper bound for cumulative standard normal distribution Ξ¦(π₯) , there are relatively not sharp for all values of the interested argument π₯ . The aim of this paper is to establish a sharp upper bound for Ξ¦(π₯) , in the sense that its maximum absolute difference from Ξ¦(π₯) is less than 5.785 Γ 10 β5 for all values of π₯ β₯ 0 . The established bound improves the well-known Polya upper bound and it can be used as an approximation for Ξ¦(π₯) itselfβ¦Β

## References

SHOWING 1-10 OF 13 REFERENCES

### Very Simple Tight Bounds on the Q-Function

- MathematicsIEEE Transactions on Communications
- 2012

New lower and upper bounds on the Gaussian Q-function are presented, unified in a single and simple algebraic expression which contains only two exponential terms with a constant and a rational coefficient, respectively, which are found to be as tight as multi-term alternatives obtained e.g. from the Exponential and Jensen-Cotes families of bounds.

### New approximations for standard normal distribution function

- MathematicsCommunications in Statistics - Theory and Methods
- 2019

Abstract This article proposes nine new approximations for the standard normal cumulative distribution function In addition, it collects most of the approximations existing in the literature. Theβ¦

### New inequalities of Mill's ratio and Its Application to The Inverse Q-function Approximation

- MathematicsArXiv
- 2012

This paper investigates the Millβs ratio estimation and gets two new inequalities and presents a conjecture on the bounds of inverse solution on Q-function and some useful results on the inverse solution.

### Inequalities and Bounds for the Incomplete Gamma Function

- Mathematics
- 2013

Inequalities involving the incomplete gamma function are established. They are obtained using logarithmic convexity of some function associated with the function in question. Lower and upper boundsβ¦

### Inequalities related to the error function

- Mathematics
- 2006

In this note we consider inequalities involving the error function $\phi$. Our methodes give new proofs of some known inequalities of Komatsu, and of Szarek and Werner, and also produce two familiesβ¦

### Bounding the Error Function

- MathematicsComputing in Science & Engineering
- 2021

Using an integral representation of the error function of a real argument, two simple and accurate lower bounds are obtained which complement a well-known upper bound given long ago by PΓ³lya.

### Probability distributions involving Gaussian random variables : a handbook for engineers and scientists

- Mathematics
- 2002

Basic Definitions and Notation.- Fundamental One-Dimensional Variables.- Fundamental Multidimensional Variables.- Difference of Chi-Square Random Variables.- Sum of Chi-Square Random Variables.-β¦

### Error function inequalities

- Computer ScienceAdv. Comput. Math.
- 2010

The inequalities for the error function are presented and one of the theorems states that Ξ±ββ₯β1 is true for all x,yβ>β0.

### New Refinements for the Error Function with Applications in Diffusion Theory

- Mathematics, Computer ScienceSymmetry
- 2020

These approximations for the error function are provided using the Pade approximation method and the Fourier series method and they are used in diffusion theory.

### Bit Error Rate Analysis for Reconfigurable Intelligent Surfaces With Phase Errors

- BusinessIEEE Communications Letters
- 2021

This letter investigates the error probability of reconfigurable intelligent surfaces (RIS)-enabled communication systems with quantized channel phase compensation with Monte Carlo simulations and derives exact and asymptotic bit error rate expressions.