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**Unformatted text preview: **X = Y almost surely. Since X is a uniform random variable on [0, 1], we know
from the previous example that E(X ) = 1/2. Therefore, using Theorem 20.3 we conclude
E(Y ) = E(X ) = 1/2. 21–3 Statistics 851 (Fall 2013)
Prof. Michael Kozdron October 28, 2013 Lecture #22: Computing Expectations of Discrete Random
Variables
Recall from introductory probability classes that a random variable X is called discrete if
the range of X is at most countable. The formula for the expectation of a discrete random
variable X given in these classes is
E(X ) = ∞
j =1 j P {X = j } . We will now derive this formula as a consequence of the general theory developed during the
past several lectures.
Suppose that the range of X is at most countable. Without loss of generality, we can assume
that Ω = {1, 2, . . . , }, F = 2Ω , and X : Ω → R is given by X (ω ) = ω . Assume further that
the law of X is given by P {X = j } = pj where pj ∈ [0, 1] and
∞
p j = 1. j =1 Let Aj = {X = j } = {ω : X (ω ) = j } so that X can be written as
X (ω ) = ∞...

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