If one has a random variable X, described by a finite probability distribution with equally likely possible values x_1,…,x_n, then the inequality: E[logX] le log(E[X]) is a reformulation of the arithmetic/geometric mean inequality.

Luisottifp 2022-09-23 Answered
Continuous/probability version of arithmetic/geometric mean inequality
If one has a random variable X, described by a finite probability distribution with equally likely possible values x 1 , , x n , then the inequality: E [ log X ] log ( E [ X ] ) is a reformulation of the arithmetic/geometric mean inequality.
And a necessary and sufficient condition for equality (maybe only if the distribution is concentrated at one point, for example)?
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Answers (1)

Rayna Aguilar
Answered 2022-09-24 Author has 14 answers
Step 1
Let X : Ω R be a random variable on any probability space ( Ω , F , μ ) such that X > 0 almost surely. Then, we have the inequality E [ log X ] log E [ X ] with equality if and only if X is constant almost surely.
Step 2
This is a special case of Jensen's inequality, E [ φ ( X ) ] φ ( E [ X ] ) which holds for convex functions φ : supp ( X ) R , where supp(X) is the smallest interval containing the essential support of X, i.e., the support of the pushforward measure induced on R by the random variable X, by setting ϕ ( x ) = log x for x ( 0 , ).
The function ϕ ( x ) = log x is strictly convex, and so by the equality conditions for Jensen's inequality, it follows that equality holds if and only if X is constant almost surely.
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