PENELOPE 2003 - OECD Nuclear Energy Agency

1.2. Random sampling methods 17
the corresponding cumulative distribution function is P(r) = 1−exp(−r 2 /2). Therefore,
r can be generated by the inverse transform method as
√
√
r = −2 ln(1 − ξ) = −2 ln ξ.
The two independent normal random variables are given by
√
x 1 = −2 ln ξ 1 cos(2πξ 2 ),
x 2 =
√
−2 ln ξ 1 sin(2πξ 2 ), (1.54)
where ξ 1 and ξ 2 are two independent random numbers. This procedure is known as the
Box-Müller method. It has the advantages of being exact and easy to program (it can
be coded as a single fortran statement).
The mean and variance of the normal variable are 〈x〉 = 0 and var(x) = 1. The
linear transformation
X = m + σx (σ > 0) (1.55)
defines a new random variable. From the properties (1.14) and (1.29), we have
The PDF of X is
〈X〉 = m and var(X) = σ 2 . (1.56)
p(X) = p G (x) dx
dX = 1 [
]
σ √ 2π exp (X − m)2
− , (1.57)
2σ 2
i.e. X is normally distributed with mean m and variance σ 2 . Hence, to generate X
we only have to sample x using the Box-Müller method and apply the transformation
(1.55).
Example 2. Uniform distribution on the unit sphere
In radiation transport, the direction of motion of a particle is described by a unit vector
ˆd. Given a certain frame of reference, the direction ˆd can be specified by giving either
its direction cosines (u, v, w) (i.e. the projections of ˆd on the directions of the coordinate
axes) or the polar angle θ and the azimuthal angle φ, defined as in fig. 1.5,
Notice that θ ∈ (0, π) and φ ∈ (0, 2π).
ˆd = (u, v, w) = (sin θ cos φ, sin θ sin φ, cos θ). (1.58)
A direction vector can be regarded as a point on the surface of the unit sphere.
Consider an isotropic source of particles, i.e. such that the initial direction (θ, φ) of
emitted particles is a random point uniformly distributed on the surface of the sphere.
The PDF is
p(θ, φ) dθ dφ = 1
[ ] [ ]
sin θ 1
4π sin θ dθ dφ = 2 dθ 2π dφ . (1.59)