Sankhya: The Indian Journal of Statistics

2000, Volume 62, Series A, Pt. 1, pp. 36--39

ON AN INEQUALITY FOR THE RAYLEIGH DISTRIBUTION

By

CHIN-YUAN HU, National Changhua University of Education, Changhua

and

GWO DONG LIN, Academia Sinica, Taipei

SUMMARY. Recently, Hitczenko (1998) gives an interesting inequality for the Rayleigh distribution, but with an incomplete proof. Specifically, let X1 and X2 be two independent random variables having common distribution F(x)=1-exp(-x2) for x>=0. Then for any two pairs (a1, a2) and $(b_1,b_2)$ of nonnegative numbers such that $(b_1^2,b_2^2)$ is majorized by $(a_1^2,a_2^2)$, P(a1X1+a2X2>=t) <=P(b1X1+b2X2>=t) for all t > 0. In this note, we offer an alternative and somewhat simpler proof for this inequality.

AMS (1991) subject classification. Primary 60E05, 60E15.

Key words and phrases. Rayleigh distribution, majorization, tail probability.

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