Continuous/probability version of arithmetic/geometric mean inequalityIf 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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Continuous/probability version of arithmetic/geometric mean inequalityIf 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)?

Rayna Aguilar · Accepted Answer

Step 1Let   X  :  Ω  →      R   be a random variable on any probability space   (  Ω  ,      F    ,  μ  ) such that   X  &amp;gt;  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 2This is a special case of Jensen&#039;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&#039;s inequality, it follows that equality holds if and only if X is constant almost surely.

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.

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