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 *X*_{1} and *X*_{2} be two independent
random variables having common distribution
F(*x*)=1-exp(-*x*^{2}) for *x*>=0. Then for any two pairs (a_{1}, a_{2})
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*(a_{1}*X*_{1}+a_{2}*X*_{2}>=*t*) <=*P*(b_{1}*X*_{1}+b_{2}X_{2}>=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.