Download Lecture 5: Expectation

Definition 5.9. A random variable X is a discrete random
variable if there exists a finite or countable set of real numbers
P
{xn } such that n pX (xn ) = 1, where pX (xn ) = P (X = xn ) = 1.
Theorem 5.5**. Let X be a discrete random variable. Then
EY = Eg(X) =
Z
Ω
g(X(ω))P (dω) =
X
n
g(xn )pX (xn ).
Examples
(1) Let Ω = {ω1 , . . . , ωN } is a discrete sample space, F = F0
is the σ-algebra of all subsets of Ω and P (A) is a probability
P
measure, which is given by the formula P (A) = ωi ∈A pi , where
p(ωi ) = P (Ai ) ≥ 0, i = 1, . . . N are probabilities of one-points
P
events Ai = {ωi } satisfying the relation ωi ∈Ω p(ωi ) = 1.
A random variable X = X(ω) and a transformed random variable Y = g(X) are, in this case, simple random variables since
P
< A1 , . . . AN > is a partition of Ω and X = ωi Ω X(ωi )IAi and
P
Y = ωi Ω g(X(ωi ))IAi .
In this case,
pX (xj ) = P (X = xj ) =
X
p(ωi )
ωi :X(ωi )=xj
and, according the definition of expectation and Theorem 1,
EY = Eg(X) =
X
g(X(ω))p(ωi ) =
ωi ∈Ω
X
n
g(xn )pX (xn ).
(2) Let Ω = {ω = (ω1 , . . . , ωn )}, ωi = 0, 1, i = 1, . . . , n} is a discrete sample space, for series of n Bernoulli trials. In this case
Q
p(ω) = ni=1 pωi q 1−ωi where p, q > 0, p + q = 1.
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