# Math Help - independent variables

1. ## independent variables

If X and Y are independent , $E(|X|^p)<\infty$ for some $p\ge1$ and E(Y)=0, then $E(|X+Y|^p)\ge E(|X|^p)$.

Any idea how to solve this problem?

Thanks.

2. Originally Posted by Veve
If X and Y are independent , $E(|X|^p)<\infty$ for some $p\ge1$ and E(Y)=0, then $E(|X+Y|^p)\ge E(|X|^p)$.

Any idea how to solve this problem?

Thanks.
This comes from the convexity of $x\mapsto |x|^p$. The curve is over its tangents, so that in particular (considering its tangent at $x$) we have for any x,y $|x+y|^p\geq |x|^p+{\rm sign}(x)p|x|^{p-1}y$. Taking expectations and using the assumption yields the result.