Does Jensen inequality, which is $\mathbb{E}(g(x)) \geq g(\mathbb{E}X)$ if $g$ is convex, assume that $\mathbb{E}X$ (expected value of random variable $X$) must belong to $R(X)$ (range of random variable $X$)?
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4$\begingroup$ No. $\phantom{a}$ $\endgroup$ShreevatsaR– ShreevatsaR2013-07-02 15:41:50 +00:00Commented Jul 2, 2013 at 15:41
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$\begingroup$ Why do you ask? $\endgroup$Did– Did2013-07-02 15:53:21 +00:00Commented Jul 2, 2013 at 15:53
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1 Answer
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The answer is negative: consider the easy example $X:\{\omega_1,\omega_2\}\rightarrow R(X)=\{x_1,x_2\}$, with $X(\omega_i)=x_i$ and $P(X=x_1)=p$, $P(X=x_2)=1-p$.
Then, for any convex $g$:
$$g(p x_1+(1-p)x_2)=g(E[X])\leq p g(x_1)+(1-p)g(x_2)=E[g(X)].$$
Note that $E[X]\not\in R(X)$, which is no linear space. It is just a finite set.
