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This paper is devoted to the study of $\Phi$-moments of sums ofindependent/freely independent random variables. More precisely, let$(f_k)_{k=1}^n$ be a sequence of positive (symmetrically distributed)independent random variables and let $\Phi$ be an Orlicz function with$\Delta_2$-condition. We provide an equivalent expression for the quantity$\mathbb{E}(\Phi(\sum_{k=1}^n f_k))$ in term of the sum of disjoint copies ofthe sequence $(f_k)_{k=1}^n.$ We also prove an analogous result in the settingof free probability. Furthermore, we provide an equivalent characterization of$\tau(\Phi(\sup^+_{1\leq k\leq n}x_k))$ for positive freely independent randomvariables and also present some new results on free Johnson-Schechtmaninequalities in the quasi-Banach symmetric operator space.

$\Phi$-moment inequalities for independent and freely independent random variables