130 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES
Xi,E[Xi],P[Xi] −→ φX(t)
y
conclusions ←− calculations
4.1 Block diagram of transform methods.
turn a difficult problem in one domain into a manageable problem in an-
other domain. Other examples are Laplace transforms, Fourier transforms,
Z-transforms, generating functions, and even logarithms.
The general method can be expressed schematically in the diagram:
Expectation of Independent Random Variables
Lemma 7. If X and Y are independent random variables, then for any
functionsgandh:
E[g(X)h(Y)] =E[g(X)]E[h(Y)]
Proof.To make the proof definite suppose thatXandY are jointly contin-
uous, with joint probability density functionF(x,y). Then:
E[g(X)h(Y)] =
∫∫
(x,y)
g(x)h(y)f(x,y)dxdy
=
∫
x
∫
y
g(x)h(y)fX(x)fY(y)dxdy
=
∫
x
g(x)fX(x)dx
∫
y
h(y)fY(y)dy
=E[g(X)]E[h(Y)].
The Moment Generating Function
Themoment generating functionφX(t) is defined by
φX(t) =E
[
etX
]
=
{∑
ie
txip(xi) ifXis discrete
∫
xe
txf(x)dx ifXis continuous
