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Chapter 1: Problem 6

Let \(X\) be a positive random variable; i.e., \(P(X \leq 0)=0\). Argue that (a) \(E(1 / X) \geq 1 / E(X)\) (b) \(E[-\log X] \geq-\log [E(X)]\) (c) \(E[\log (1 / X)] \geq \log [1 / E(X)]\) (d) \(E\left[X^{3}\right] \geq[E(X)]^{3}\).

Short Answer

Expert verified
Each part of the exercise can be solved by applying Jensen's inequality to an appropriately chosen convex function. For part (a), \(f(x) = 1/x\). For part (b), \(f(x) = -\log(x)\). For part (c), \(f(x) = \log(1/x)\). For part (d), \(f(x) = x^3\). The constraint that \(X\) is a positive random variable is required for the chosen functions to be convex.

Step by step solution

01

Using Jensen's Inequality for Part (a)

Consider the function \(f(x) = 1/x\) which is convex for \(x > 0\). Since \(f\) is convex and \(X\) is a positive random variable, we can apply Jensen's inequality to obtain: \[E[1 / X] \geq 1 / E[X]\].
02

Using Jensen's Inequality for Part (b)

Consider the function \(f(x) = -\log(x)\) which is convex for \(x > 0\). Similar to step 1, we can apply Jensen's inequality to obtain: \[E[-\log X] \geq-\log [E(X)]\].
03

Using Jensen's Inequality for Part (c)

In this part, consider the function \(f(x) = \log(1/x) = -\log(x)\). Just like in previous steps, Jensen's inequality can be applied: \[E[\log (1 / X)] \geq \log [1 / E(X)]\].
04

Using Jensen's Inequality for Part (d)

For the final part, consider \(f(x) = x^{3}\) which is a convex function for \(x > 0\). Applying Jensen's inequality like in previous steps, we get \[E\left[X^{3}\right] \geq[E(X)]^{3}\].

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