ABSTRACT

The cdf can be expressed in terms of the cdf of the standard normal distribution as

F (x; 1 , 0') = 2<1> ( ~) - l , (9.2.3) where <I>() is the cdf of the standard normal distribution (0, 1).

9.2.2 The Two-Dimensional Rayleigh Distribution, p = 2 This special case is perhaps the most important of all the Rayleigh distributions. It is the one that received the most attention from its discoverer, and may users are inclined to consider it as the only Rayleigh distribution. It is not only a special case of the general Rayleigh distribution, it is also a special case of the Weibull distribution. In connection with applications concerning the analysis of target errors, it is sometimes referred to as the circular normal distribution

With p = 2, the pdf of (9.1.2) becomes

f(x; 2, 0') = xz exp (- x2z), (J' 20' 0 <X< oo, (9.2.4)

This same pdf follows from the two-parameter Weibull distribution (~. 8) with pdf (5.2.1) when we set 3 = 2 and ~2 = 2<T2. Moments and related properties of this distribution can be obtained from either the Weibull properties or the pdimensional Rayleigh properties. The Weibull parameterization (~. 2) enables the pdf to be written as

2x ( x2) j(x; 2, ~) = ~2 exp - ~2 , O<x<oo (9.2.5) = 0 elsewhere,

or with fl = ~2 , as

Thus, we have three equivalent parameterizations for this distribution. The cdf with <T as the parameter is

F(x; 2, <T) = l - exp (- 2: 2). (9.2.7)

The kth moment about the origin is

V(X) = 2 = 0.429204<J2,

2(1T - 3) V1T a3(X) = (4 _ 1r)312 = 0.631110, (9.2.9)

32 - 31T2 aiX) = (4 2 = 3.245089,

The rth percentile is

(9.2.10)

and the hazard function h(x) = f(x)l[ 1 - F(x)], which is of interest in reliability analysis, is the increasing function

a3(X) = ( 31r _ 8)312 = 0.485693,

1511" 2 + 1611" - 192 a 4(X) = (311" _ W = 3.108164,

(9.2.11)

(9.2.12)

(9.2.13)

This distribution is of special interest to engineers and physicists. It was originally derived by Maxwell (1860) as the distribution of the velocity of gas particles in three-dimensional space. Boltzmann (1877) used Maxwell's results in explaining the thermodynamics of gases on the basis of kinetic theory in which gases were viewed as particles undergoing movement at different velocities and

collisions according to the principles of mechanics. In recognition of his contributions, Boltzmann's name has also become associated with this distribution.