Consider approximating the normal tail integral

\[\bar{\Phi}(z)=\int_{z}^{\infty} \phi(t) d t\]

for large values of \(z\) using integration by parts as discussed in Example 1.24. Use repeated integration by parts to show that

\[\bar{\Phi}(z)=z^{-1} \phi(z)-z^{-3} \phi(z)+3 z^{-5} \phi(z)-15 z^{-7} \phi(z)+O\left(z^{-9}ight),\]as \(z ightarrow \infty\).

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Example 1.24. Consider the problem of approximating the tail probability function of a standard normal distribution given by (z) = (t)dt, 2 for large values of 2, or as 200. To apply integration by parts, first note that from Definition 1.6 it follows that o' (t) = -H(t)o(t) = -to(t) or equivalently o(t)='(t)/t. Therefore (2) - -foot-b' (t) dt.