Let $X$, $Y$ be independent random variables, $E\left(\left|X\right|^p\right)<+\infty$ where $p\geq 1$ and $E(Y)=0$. Show that $E\left(\left|X+Y\right|^p\right)\geq E(\left|X\right|^p)$, where $E\left(\cdot\right)$ stands for expectation.
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Hint: for every fixed $x$ and every random variable $Y$, $E(|x+Y|^p)≥|x+E(Y)|^p$.
Jensen's inequality seems to be the way to prove this, hence the first goal is to find a convex function somewhere... Once you know this, the rest should be easy.