Let \(\psi_{0}(y)=e^{-y^{2} / 2} / \sqrt{2 \pi}\) be the standard normal density. Assume that \(Y_{1}, \ldots, Y_{n}\) are independent standard normal. Show that the random variables \(X_{i}=\) \(\psi_{1}\left(Y_{i}ight) / \psi_{0}\left(Y_{i}ight)\) have unit mean, and hence, by the law of large numbers, that the sample average tends to one as \(n ightarrow \infty\).