
28 Modeling and In verse Problems in the Presence of Uncertainty
which implies that the characteristic function Π is essentially the Fourier
transform of the probability density function p of X. Hence, p can be uniquely
determined from the inverse Fourier transform of its associated characteristic
function and is given by
p(x) =
1
2π
Z
∞
−∞
exp (− iςx) Π(ς)dς. (2.70)
Interested readers are referred to some popular books such as [5, 16, 19] for
more info rmation on the Fourier transform and its applications.
The characteristic function has many useful properties. For example, Π is
bounded by Π(0) = 1, which can be dire ctly deduced from (2.69). In addition, ...