Suppose X is a non-negative random variable whose mean exists. Consider: (I) E(2log X)le 2E(log X) (II) E(exp(-2X)) ge exp(-2E(X)) (III) E(X^1/5) ge (E(X))^1/5

Q112Section III (Statistics)112 of 180

Jensen's inequality — direction of the inequality

Suppose $X$ is a non-negative random variable whose mean exists. Consider:

(I) $\mathbb{E}(2\log X)\le 2\mathbb{E}(\log X)$
(II) $\mathbb{E}\bigl(\exp(-2X)\bigr) \ge \exp\bigl(-2\mathbb{E}(X)\bigr)$
(III) $\mathbb{E}\bigl(X^{1/5}\bigr) \ge \bigl(\mathbb{E}(X)\bigr)^{1/5}$

Pick an option to check your answer.

Keyboard: A–D answer · S solution · ←/→ previous/next