Question

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

For (a), (b), (c), and (d), Jensen's inequality is used to establish the respective inequality. As all the functions used in (a) to \(f(x) = 1/x\), (b) \(f(x) = -\log x\), (c) \(f(x) = \log(1/x)\), and (d) \(f(x) = x^3\) are convex, the inequality holds true for every case as all operations are performed on positive random variable \(X\).

Step-by-step solution

Problem (a) \(E(1 / X) \geq 1 / E(X)\)

Let \(f(x) = 1/x\). Since \(X\) is a positive random variable, \(f(x)\) is a convex function on \(X\). So, we use Jensen's inequality (which states that the expectation of a convex function of a random variable is always greater than or equal to the function of its expectation) to obtain this inequality: \(E(1 / X) \geq 1 / E(X)\).

Problem (b) \(E[-\log X] \geq-\log [E(X)]\)

Now, let \(f(x) = -\log x\). Again, as \(X\) is a positive random variable, \(f(x)\) is a convex function on \(X\). Using Jensen's inequality again, we find: \(E[-\log X] \geq -\log [E(X)]\).

Problem (c) \(E[\log (1 / X)] \geq \log [1 / E(X)]\)

Here, let \(f(x) = \log(1/x)\). Since \(X\) is a positive random variable, \(f(x)\) is a convex function on \(X\). Applying once again Jensen's inequality, we obtain: \(E[\log(1/X)] \geq \log[1/E(X)]\).

Problem (d) \(E[X^3] \geq [E(X)]^3\)

For this, let \(f(x) = x^3\). This function is convex for positive \(X\). Using Jensen's inequality for the final time, we find: \(E[X^3] \geq [E(X)]^3\).