Modeling and Inverse Problems in the Presence of Uncertainty

Probability and Statistics Overview 13

2.2.3 Probability Density Function

If the derivative p of the cumulative distribution function P exists for almost

all x, and for all x

P (x) =

Z

x

−∞

p(ξ)dξ,

then p is called the probability density function. It should be noted that a

necessary and sufficient condition for a cumulative distribution function to

have an associated probability density function is that P is absolutely con-

tinuous in the sense that for any positive number ǫ, there exists a positive

number c

l

such that for any finite collectio n of disjoint intervals (x

j

, y

j

) ⊂ R

satisfying

X

j

|y

j

− x

j

| < c

l

then

X

j

|P (y

j

) − P (x

j

)| < ǫ. Thus, we se e that

the requirement for absolute continuity is much stronger than that for conti-

nuity. This implies tha t there ...