In other words, does $$ E\big[E[f(X)]\big] \ge E\big[f(E[X])\big] $$ holds for a convex function $f(x)$?
If so why? or why not?
2026-03-26 07:32:28.1774510348
Does the Jensen's inequality hold under expectation?
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By Jensen, $E[f(X)] >= f(E[X])$. Therefore
$$E[E[f(X)]] = E[f(X)] ≥ f(E[X]) = E[f(E[X])$$
is true.
Any true inequality holds under expectation.
$A\geq B \Rightarrow E_X[A]=A\geq B=E_X[B]$