91Ó°ÊÓ

X is a nonnegative random variable with a mean$25$,

Suppose that $X$is a non-negative random variable$(Xâ‰\yen 0)$with a mean$\mathrm{μ}=E\left[X\right]=25$.

$\left(a\right)$The function $f\left(x\right)=x^3$is convex function f$xâ‰\yen 0$. By Jensen's inequality,

role="math" localid="1649837548313" $$\begin{gathered} E\left[f\right(X\left)\right]â‰\yen f\left(E\right[X\left]\right)⇔E\left[X^3\right]â‰\yen \left(E\right[X]{)}^3={\mathrm{μ}}^3=15625 \\ E\left[X^3\right]â‰\yen 15625 \end{gathered}$$

$\left(b\right)$The function localid="1649838581216" $f\left(x\right)=-\sqrt{x}$is a convex function$xâ‰\yen 0$. By Jensen's inequality,

$$\begin{gathered} E[-\sqrt{X}]â‰\yen -\sqrt{E\left[X\right]}⇔E\left[\sqrt{X}\right]≤\sqrt{E\left[X\right]}=\sqrt{\mathrm{μ}}=5 \\ E\left[\sqrt{X}\right]≤5 \end{gathered}$$

$\left(c\right)$The function localid="1649838550535" $f\left(x\right)=-\log x$is a convex functionlocalid="1649838558394" $xâ‰\yen 0$. By Jensen's inequality,

$$\begin{gathered} E[-\log X]â‰\yen -\log E\left[X\right]⇔E\left[\log X\right]≤\log E\left[X\right]=\log \mathrm{μ}≈1.3979 \\ E\left[\log X\right]≤1.3979 \end{gathered}$$

$\left(d\right)$The function $f\left(x\right)=e^{-x}$is a convex function$xâ‰\yen 0$. By Jensen's inequality,

$$\begin{gathered} E\left[e^{-X}\right]â‰\yen e^{-E\left[X\right]}=e^{-\mathrm{μ}}=e^{-25} \\ E\left[e^{-X}\right]â‰\yen e^{-25} \end{gathered}$$