Let ((X, Y)) be a mean zero Gaussian random variable. Show that (mathbb{E}left(X^{2} Yight)=0). Use this observation

Question:

Let \((X, Y)\) be a mean zero Gaussian random variable. Show that \(\mathbb{E}\left(X^{2} Yight)=0\). Use this observation to show that \(I^{2}(f)-\|f\|_{L^{2}}^{2}\) is and \(\mathcal{K}_{1}:=\left\{I_{1}(g): g \in L^{2}\left(\mathbb{R}_{+}ight)ight\}\)are orthogonal.

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