Q. 8.20 - A First Course in Probability

Question

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

$\left(a\right) E\left[X^3\right]?$

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

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

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

Step-by-Step Solution

Answer

$\left(a\right)$ width="105" height="22" style="max-width: none; vertical-align: -4px;" $E\left[X^3\right]\ge 15625$

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

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

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

Step 1 Given Information

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

Step 2 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,

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

Step 3 part (b) Explanation.

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

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

Step 4 part (c) Explanation.

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

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

Step 5 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,

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