Prove that $\mathrm{E}(X \ln X) \ge \mathrm{E}(X) \mathrm{E}(\ln X)$.
Can I use Jensen's inequality? Should I prove that $g(x)=x\ln x$ is a convex function? I don't know how to prove it?
2026-03-26 09:36:48.1774517808
Prove that $\mathrm{E}(X \ln X) \ge \mathrm{E}(X) \mathrm{E}(\ln X)$
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$x\,ln (x)$ is a convex function on $(0,\infty)$since its second derivative $\frac 1 x$ is positive. The inequality simply says $f(EX)\leq Ef(X)$ which is Jensen's inequality.