PENELOPE 2003 - OECD Nuclear Energy Agency

1.1. Elements of probability theory 3
which is discontinuous. The definition (1.2) also includes singular distributions such as
the Dirac delta, δ(x − x 0 ), which is defined by the property
⎧
∫ b
⎨ f(x 0 ) if a < x 0 < b,
f(x)δ(x − x 0 ) dx =
(1.4)
a
⎩
0 if x 0 < a or x 0 > b
for any function f(x) that is continuous at x 0 . An equivalent, more intuitive definition
is the following,
δ(x − x 0 ) ≡ lim U x0 −∆,x 0 ∆→0
+∆(x), (1.4 ′ )
which represents the delta distribution as the zero-width limit of a sequence of uniform
distributions centred at the point x 0 . Hence, the Dirac distribution describes a singlevalued
discrete random variable (i.e. a constant). The PDF of a random variable x
that takes the discrete values x = x 1 , x 2 , . . . with point probabilities p 1 , p 2 , . . . can be
expressed as a mixture of delta distributions,
p(x) = ∑ i
p i δ(x − x i ). (1.5)
Discrete distributions can thus be regarded as particular forms of continuous distributions.
Given a continuous random variable x, the cumulative distribution function of x is
defined by
∫ x
P(x) ≡ p(x ′ ) dx ′ . (1.6)
x min
This is a non-decreasing function of x that varies from P(x min ) = 0 to P(x max ) = 1. In
the case of a discrete PDF of the form (1.5), P(x) is a step function. Notice that the
probability P{x|a < x < b} of having x in the interval (a,b) is
and that p(x) = dP(x)/dx.
P{x| a < x < b } =
The n-th moment of p(x) is defined as
〈x n 〉 =
∫ b
a
∫ xmax
p(x) dx = P(b) − P(a), (1.7)
x min
x n p(x) dx. (1.8)
The moment 〈x 0 〉 is simply the integral of p(x), which is equal to unity, by definition.
However, higher order moments may or may not exist. An example of a PDF that has
no even-order moments is the Lorentz or Cauchy distribution,
p L (x) ≡ 1 π
γ
γ 2 + x2, −∞ < x < ∞. (1.9)
Its first moment, and other odd-order moments, can be assigned a finite value if they
are defined as the “principal value” of the integrals, e.g.
〈x〉 L = lim
a→∞
∫ +a
−a
x 1 π
γ
dx = 0, (1.10)
γ 2 + x2