Discussion
Suppose X and Y are two random variables whose third order moments exist. Then \((E(|X + Y|^{3}))^{1/3}\) is less than or equal to :
Answer: B
The bound is \((E|X|^{3})^{1/3} + (E|Y|^{3})^{1/3}\).
This is Minkowski's inequality in \(L^{3}\): the \(L^{p}\) norm \(\|Z\|_{p} = (E|Z|^{p})^{1/p}\) satisfies the triangle inequality for \(p \ge 1\), so
\[ \|X + Y\|_{3} \le \|X\|_{3} + \|Y\|_{3} \]
The product form (A) fails, for example with \(Y = 0\): the left side is \(\|X\|_{3}\) but the right side is 0. The forms with first absolute moments do not hold in general either, since \(L^{3}\) norms can be much larger than \(L^{1}\) ones.
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