An Exponential Inequality for Symmetric Random Variables

We prove the following exponential inequality: Let $n\ge 1$ and let $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n$ be $n$ independent identically distributed symmetric real-valued random variables. For any $x,y>0$, we have $$\mathbb{P}(X_1+\mathrm{.}\mathrm{.}\mathrm{.}+X_n\ge x,X_1^2+\mathrm{.}\mathrm{.}\mathrm{.}+X_n^2\le y)<\exp (-\frac{x^2}{2y})\mathrm{.}$$