If $X$ is a non negative random variable with $E[X]=25$, what can be said about $E[X^3]$, $E[\sqrt{X}]$, $E[\ln X]$, and $E[e^{-X}]$?

Question

How I solve this? (Jensen's inequality?)

Answer 1

Jensen's inequality

\begin{align} f \text{ concave}:& \;\;\; E[f(X)] \leq f(E[X]) \\[2ex] f \text{ convex}:& \;\;\; E[f(X)] \geq f(E[X]) \end{align}

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Since $f(X)=\sqrt{X}$ and $f(X)=\ln{X}$ are concave, it follows that \begin{align} E\left[\sqrt{X}\right] &\leq \sqrt{E[X]} = \sqrt{25} \\[2ex] E\left[\ln{X}\right] &\leq \ln E[X] = \ln 25 \end{align}

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Since $f(X)=X^3$ and $f(X)=e^{-X}$ are convex, it follows that \begin{align} E\left[X^3\right] &\geq E[X]^3 = 25^3 \\[2ex] E\left[e^{-X}\right] &\geq e^{-E[X]} = e^{-25} \end{align}