Page 332 - Elements of Distribution Theory

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052184472Xc10 CUNY148/Severini May 24, 2005 2:50

318 Orthogonal Polynomials

λ 1 ,λ 2 ,...,λ n such that if f is a polynomial of degree 2n − 1or less,

b n

f (x) dF(x) = λ j f (x j ).
a j=1
Now suppose that f is not a polynomial, but may be approximated by a polynomial.
Write
2n−1

j
f (x) = β j x + R 2n (x)
j=0
where β 0 ,...,β 2n−1 are given constants and R 2n is a remainder term. Then
b b b
f (x) dF(x) = f ˆ (x) dF(x) + R 2n (x) dF(x)
2n−1
a a a
where
2n−1
j

f ˆ (x) = β j x .
2n−1
j=0
Since f ˆ is a polynomial of degree 2n − 1, it may be integrated exactly:
2n−1
n
b
f ˆ (x) dF(x) = λ j f ˆ (x j ).
2n−1 2n−1
a j=1
The function f ˆ may be approximated by f so that
2n−1
n
b .
f ˆ 2n−1 (x) dF(x) = λ j f (x j )
a j=1
and, provided that
b
R 2n (x) dF(x)
a
is small,
b n
.
f (x) dF(x) = λ j f (x j ).
a j=1
This approach to numerical integration is known as Gaussian quadrature. Clearly, this
method will work well whenever the function f being integrated can be approximated
accurately by a polynomial over the range of integration.
In order to use Gaussian quadrature, we need to know the zeros of an orthogonal poly-
nomial p n , along with the corresponding weights λ 1 ,...,λ n . There are many published
sources containing this information, as well as computer programs for this purpose; see
Section 10.6 for further details.

Example 10.11. Let X denote a random variable with a standard exponential distribution
and consider computation of E[g(X)] for various functions g. Since
∞

E[g(X)] = g(x)exp(−x) dx,
0