Download PDF

generalized mean∗
Mathprof†
2013-03-21 17:44:45
Definition
Let x1 , x2 , . . . , xn be real numbers, and f a continuous and strictly increasing
or decreasing function
Pn on the real numbers. If each number xi is assigned a
weight pi , with
i=1 pi = 1, for i = 1, . . . , n, then the generalized mean is
defined as
n
X
f −1
pi f (xi ) .
i=1
Special cases
1. f (x) = x, pi = 1/n for all i: arithmetic mean
2. f (x) = x: weighted mean
3. f (x) = log(x), pi = 1/n for all i: geometric mean
4. f (x) = 1/x and pi = 1/n for all i: harmonic mean
5. f (x) = x2 and pi = 1/n for all i: root-mean-square
6. f (x) = xd and pi = 1/n for all i: power mean
7. f (x) = xd : weighted power mean
8. f (x) = 2(1−α)x , α 6= 1, xi = − log2 pi : Rényi’s α-entropy
∗ hGeneralizedMeani
created: h2013-03-21i by: hMathprofi version: h36081i Privacy
setting: h1i hDefinitioni h26-00i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1