Question: Let (X sim mathrm{N}(0,1)). Show that for every (lambda_{0} inleft(0, frac{1}{2}ight)) there is a constant (C=Cleft(lambda_{0}ight)) such that (sup _{lambda leqslant lambda_{0}} mathbb{E}left(left(X^{2}-1ight)^{2} e^{|lambda|left(X^{2}-1ight)}ight) leqslant

Let $X \sim \mathrm{N}(0,1)$. Show that for every $\lambda_{0} \in\left(0, \frac{1}{2}ight)$ there is a constant $C=C\left(\lambda_{0}ight)$ such that $\sup _{\lambda \leqslant \lambda_{0}} \mathbb{E}\left(\left(X^{2}-1ight)^{2} e^{|\lambda|\left(X^{2}-1ight)}ight) \leqslant C<\infty$.

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