Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / F / F2 / c / 2

Table of contents
- Solution 2

Solution

A = \sum_{i} p_{i} a_{i} > 0

and

x_{i} = a_{i} / A

. Then

\sum_{i} p_{i} x_{i} = 1

, so part 1 gives

\prod_{i} (a_{i} / A)^{p_{i}} \leq 1

. Since

\sum_{i} p_{i} = 1

, this becomes

i \prod a_{i}^{p_{i}} \leq i \sum p_{i} a_{i} .

Equality holds exactly when

x_{i} = 1

for all

i

, equivalently all the

a_{i}

are equal. This is the weighted arithmetic–geometric mean inequality.

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