(See Steps) Chebyshev's other inequality. Let f: R \rightarrow R and g: R \rightarrow R be bounded and increasing functions. Prove that, for any r.v. X, E(f(X)

Question: Chebyshev's other inequality.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be bounded and increasing functions. Prove that, for any r.v. $X$,

\[E(f(X) g(X)) \geq(E f(X))(E g(X))\]

[In other words, $f(X)$ and $g(X)$ are positively correlated. This is intuitively obvious, but a little tricky to prove. Hint: consider an independent copy $Y$ of $X$. For this and the next question you may need the product rule for expectations of independent r.v.s]

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(See Steps) Chebyshev's other inequality. Let f: R \rightarrow R and g: R \rightarrow R be bounded and increasing functions. Prove that, for any r.v. X, E(f(X)

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