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generalized mean
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(Definition)
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Definition
Let $x_1$ , $x_2,\ldots, x_n$ be real numbers, and $f$ a continuous and strictly increasing or decreasing function on the real numbers. If each number $x_i$ is assigned a weight $p_i$ , with
$\sum_{i=1}^n p_i= 1$ , for $i=1,\ldots,n$ , then the generalized mean is defined as $$ f^{-1}\Big( \sum_{i=1}^n p_i f(x_i) \Big). $$
Special cases
- $f(x)=x$ , $p_i=1/n$ for all $i$ : arithmetic mean
- $f(x)=x$ : weighted mean
- $f(x)=\log(x)$ , $p_i=1/n$ for all $i$ : geometric mean
- $f(x)=1/x$ and $p_i=1/n$ for all $i$ : harmonic mean
- $f(x)=x^2$ and $p_i=1/n$ for all $i$ : root-mean-square
- $f(x)=x^d$ and $p_i=1/n$ for all $i$ : power mean
- $f(x)=x^d$ : weighted power mean
- $f(x)=2^{(1-\alpha)x}$ , $\alpha\neq 1$ , $x_i=-\log_2 p_i$ : Rényi's $\alpha$ -entropy
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"generalized mean" is owned by Mathprof. [ full author list (2) | owner history (1) ]
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| Other names: |
Kolmogorov-Nagumo function of the mean, Hölder mean |
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Cross-references: weighted power mean, power mean, root-mean-square, harmonic mean, geometric mean, weighted mean, arithmetic mean, weight, number, function, decreasing, strictly increasing, continuous, real numbers
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This is version 5 of generalized mean, born on 2004-08-06, modified 2006-10-24.
Object id is 6081, canonical name is GeneralizedMean.
Accessed 4710 times total.
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Pending Errata and Addenda
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