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Rayleigh distribution (RD) has wide applications in many real life
situations especially life testing, reliability analysis, medicines etc. In
this paper a new weighted Rayleigh distribution named area-biased Rayleigh
distribution (ARD) is introduced. Some mathematical properties of the (ARD)
including cumulative distribution function, moments, skewness, kurtosis, median,
mode, entropy, reliability measures as survival function and hazard function
have been derived. Parameter of the ARD is estimated by method of moments
(MOM), maximum likelihood (ML), and Bayesian.
Properties of the estimators are developed. It is proved that the ML estimator attains
the Cramer Rao lower bound. Applications of the ARD provided for some life time
data sets. Kolmogorov Smirnov (K-S) test statistics is applied to check the
good fit of the model.
This paper studies the global behavior defocusing nonlinear
Schrodinger equation in dimension d = 2, and we will discuss the case . This means that the
solutions , and called critical solution. We show that u scatters forward and backward to a free
solution and the solution is globally well posed.