Question

If, $X$is a nonnegative random variable with a mean of$25$, what can be said about

role="math" localid="1649836663727" $\left(a\right)E\left[X^3\right]?$

$\left(b\right)E[\surd X]?$

$\left(c\right)E[\log X]?$

role="math" localid="1649836643047" $\left(d\right)E\left[e^{-X}\right]?$

Short answer

$\left(a\right)$width="105" height="22" role="math">EX3≥15625

$\left(b\right)$$E\left[\sqrt{X}\right]\le 5$

$\left(c\right)$$E\left[\log X\right]\le 1.3979$

$\left(d\right)$$E\left[e^{-X}\right]\ge e^{-25}$

Step-by-step solution

Given Information

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

part (a) Explanation.

Suppose that $X$is a non-negative random variable$(X\ge 0)$with a mean$\mu =E\left[X\right]=25$.

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

role="math" localid="1649837548313" $$\begin{gathered} E\left[f\right(X\left)\right]\ge f\left(E\right[X\left]\right)\iff E\left[X^3\right]\ge \left(E\right[X]{)}^3={\mu }^3=15625 \\ E\left[X^3\right]\ge 15625 \end{gathered}$$

part (b) Explanation.

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

$$\begin{gathered} E[-\sqrt{X}]\ge -\sqrt{E\left[X\right]}\iff E\left[\sqrt{X}\right]\le \sqrt{E\left[X\right]}=\sqrt{\mu }=5 \\ E\left[\sqrt{X}\right]\le 5 \end{gathered}$$

part (c) Explanation.

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

$$\begin{gathered} E[-\log X]\ge -\log E\left[X\right]\iff E\left[\log X\right]\le \log E\left[X\right]=\log \mu \approx 1.3979 \\ E\left[\log X\right]\le 1.3979 \end{gathered}$$

part (d) Explanation.

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

$$\begin{gathered} E\left[e^{-X}\right]\ge e^{-E\left[X\right]}=e^{-\mu }=e^{-25} \\ E\left[e^{-X}\right]\ge e^{-25} \end{gathered}$$