Download Lecture 5: Expectation

Definition 5.8 If function f is bounded and Riemann-Stiltjes
Rn
integrable on any finite interval, and limn→∞ −n
|f (x)|dG(x) <
∞, then function f is Riemann-Stiltjes integrable on real line
R∞
Rn
and −∞
f (x)dG(x) = limn→∞ −n
f (x)dG(x).
Theorem 5.3*. A real-valued bounded Borel function f (x)
defined on a real line is Riemann-Stiltjes integrable on [a, b] if
and only if its set of discontinuity points Rf [a, b] has the measure G(Rf [a, b]) = 0.
Theorem 5.4*. If Ω = R1 , and F = B1 and f = f (x) is a
R∞
Riemann-Stiltjes integrable function, i.e., −∞
|f (x)|dG(x) < ∞.
R∞
Then the Lebesgue integral −∞ |f (x)|G(dx) < ∞ and
Z ∞
−∞
f (x)dG(x) =
Z ∞
−∞
f (x)G(dx).
2. Expectation and distribution of random variables
2.1 Expectation for transformed discrete random variables
< Ω, F, P > is a probability space;
X = X(ω) is a real valued random variable defined on the probability space < Ω, F, P >.
g(x) is a Borel real-valued function defined on a real line.
Y = g(X) is a transformed random variable.
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