Chapter 5 Discrete Distributions

5.4. EXPECTATION AND MOMENT GENERATING FUNCTIONS 117
Definition 5.10. More generally, given a function g we define the expected value of g(X) by
This should be a theorem, ∑
not a definition IE g(X) = g(x) f X (x), (5.4.2)
x∈S
provided the (potentially infinite) series ∑ x |g(x)| f (x)isconvergent.WesaythatIEg(X) exists.
In this notation the variance is σ 2 = IE(X − µ) 2 and we prove the identity
IE(X − µ) 2 = IE X 2 − (IE X) 2 (5.4.3)
in Exercise 5.4. Intuitively,forrepeatedobservationsofX we would expect the sample mean
of the g(X)valuestocloselyapproximateIEg(X) asthesamplesizeincreaseswithoutbound.
Let us take the analogy further. If we expect g(X) tobeclosetoIEg(X) ontheaverage,
where would we expect 3g(X) tobeontheaverageItcouldonlybe3IEg(X). The following
theorem makes this idea precise.
Proposition 5.11. For any functions g and h, any random variable X, and any constant c:
1. IE c = c,
2. IE[c · g(X)] = c IE g(X)
3. IE[g(X) + h(X)] = IE g(X) + IE h(X),
provided IE g(X) and IE h(X) exist.
Proof. Go directly from the definition. For example,
∑
∑
IE[c · g(X)] = c · g(x) f X (x) = c · g(x) f X (x) = c IE g(X).
x∈S
x∈S
□
5.4.2 Moment Generating Functions
Definition 5.12. Given a random variable X, itsmoment generating function (abbreviated
MGF) is defined by the formula
∑
M X (t) = IE e tX = e tx f X (x), (5.4.4)
provided the (potentially infinite) series is convergent for allt in a neighborhood of zero (that
is, for all −ɛ < t < ɛ, forsomeɛ > 0).
Note that for any MGF M X ,
x∈S
M X (0) = IE e 0·X = IE 1 = 1. (5.4.5)
We will calculate the MGF for the two distributions introduced above.
Example 5.13. Find the MGF for X ∼ disunif(m).
Since f (x) = 1/m, theMGFtakestheform
M(t) =
m∑
e tx 1 m = 1 m (et + e 2t + ···+ e mt ), for any t.
x=1