Slides Chapter 1. Measure Theory and Probability

1.8. RANDOM VARIABLES
(b) F X absolutely continuous: In this case,
∫ ∫
gdP X = g ·f X dµ
IR
and if g · f X is Riemann integrable, then it is also Lebesgue
integrable and the two integrals coincide, i.e.,
∫ ∫ ∫ +∞
gdP X = g ·f X dµ = g(x)f X (x)dx.
IR
IR
Definition 1.47 The expectation of the r.v. X is defined as
∫
µ = E(X) = XdP.
IR
IR
−∞
Corolary 1.2 The expectation of X can be calculated as
∫
E(X) = xdF X (x),
IR
and provided that X is absolutely continuous with p.d.f. f X (x),
then
∫
E(X) = xf X (x)dx.
IR
Definition 1.48 The k-th moment of X with respect to a ∈ IR
is defined as
α k,a = E(g k,a ◦X),
where g k,a (x) = (x−a) k , provided that the expectation exists.
Remark 1.10 It holds that
∫ ∫ ∫
α k,a = g k,a ◦XdP = g k,a (x)dF X (x) =
IR
IR
IR
(x−a) k dF X (x).
Observe that for the calculation of the moments of a r.v. X we
only require is its d.f.
ISABEL MOLINA 43