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2. Expectations of Functions of Random Variables 77

Then, we express E(X ) analogously as
2

2
In other words, E(X ) simplifies to

2
2
2
Hence, V(X) = E(X ) – E (X) = (α + 1)αβ – α β = αβ . In summary, for
2 2
2
the random variable X distributed as Gamma(α, β), we have
2.3 The Moments and Moment Generating
Function

Start with a random variable X and consider a function of the random variable,
g(X). Suppose that f(x) is the pmf or pdf of X where x ∈ χ is the support of the
distribution of X. Now, recall the Definition 2.2.3 for the expected value of the
random variable g(X), denoted by E[g(X)].
We continue to write µ = E(X). When we specialize g(x) = x – µ, we obvi-
ously get E[g(X)] = 0. Next, if we let g(x) = (x – µ) , we get E[g(X)] = σ . Now
2
2
these notions are further extended by considering two other special choices of
r
functions, namely, g(x) = x or (x – µ) for fixed r = 1, 2, ... .
r
th
Definition 2.3.1 The r moment of a random variable X, denoted by η , is
r
given by η = E[X ], for fixed r = 1, 2, ... . The first moment η is the mean or
r
r
1
the expected value µ of the random variable X.
Definition 2.3.2 The r central moment of a random variable X around its
th
mean µ, denoted by µ , is given by µ = E[(X – µ) ] with fixed r = 1, 2, ... .
r
r r
Recall that the first central moment µ is zero and the second
1
2
central moment µ turns out to be the variance σ of X,
2
assuming that µ and σ are finite.
2
Example 2.3.1 It is known that the infinite series converges
if p > 1, and it diverges if p ≤ 1. Refer back to (1.6.12). With some fixed