Chapter 8: Q. 8.20 (page 391)

If, $X$ is a nonnegative random variable with a mean of $25$ , what can be said about
role="math" localid="1649836663727" $(a) E [X^{3}] ?$
$(b) E [\sqrt X] ?$
$(c) E [\log X] ?$
role="math" localid="1649836643047" $(d) E [e^{- X}] ?$

Short Answer

Expert verified

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

$(b)$ $E [\sqrt{X}] \leq 5$

$(c)$ $E [\log X] \leq 1.3979$

$(d)$ $E [e^{- X}] \geq e^{- 25}$

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 $(X \geq 0)$ with a mean $μ = E [X] = 25$ .

$(a)$ The function $f (x) = x^{3}$ is convex function f $x \geq 0$ . By Jensen's inequality,

role="math" localid="1649837548313" $E [f (X)] \geq f (E [X]) \Leftrightarrow E [X^{3}] \geq (E [X])^{3} = μ^{3} = 15625 E [X^{3}] \geq 15625$

part (b) Explanation.

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

$E [- \sqrt{X}] \geq - \sqrt{E [X]} \Leftrightarrow E [\sqrt{X}] \leq \sqrt{E [X]} = \sqrt{μ} = 5 E [\sqrt{X}] \leq 5$

part (c) Explanation.

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

$E [- \log X] \geq - \log E [X] \Leftrightarrow E [\log X] \leq \log E [X] = \log μ \approx 1.3979 E [\log X] \leq 1.3979$

part (d) Explanation.

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

$E [e^{- X}] \geq e^{- E [X]} = e^{- μ} = e^{- 25} E [e^{- X}] \geq e^{- 25}$

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If, $X$ is a nonnegative random variable with a mean of $25$ , what can be said about
role="math" localid="1649836663727" $(a) E [X^{3}] ?$
$(b) E [\sqrt X] ?$
$(c) E [\log X] ?$
role="math" localid="1649836643047" $(d) E [e^{- X}] ?$

Short Answer

Step by step solution

Given Information

part (a) Explanation.

part (b) Explanation.

part (c) Explanation.

part (d) Explanation.

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If, Xis a nonnegative random variable with a mean of25, what can be said aboutrole="math" localid="1649836663727" (a)E[X3]?(b)E[√X]?(c)E[logX]?role="math" localid="1649836643047" (d)E[e-X]?

Short Answer

Step by step solution

Given Information

part (a) Explanation.

part (b) Explanation.

part (c) Explanation.

part (d) Explanation.

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Most popular questions from this chapter

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Theoretical and Mathematical Physics

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If, $X$ is a nonnegative random variable with a mean of $25$ , what can be said about
role="math" localid="1649836663727" $(a) E [X^{3}] ?$
$(b) E [\sqrt X] ?$
$(c) E [\log X] ?$
role="math" localid="1649836643047" $(d) E [e^{- X}] ?$