Let X be a standard normal random variable and let \(\sigma\) be a positive random variable independent of X. The distribution of \(\eta =\sigma X\) is expanded around that of N(0,1) and its error bounds are obtained. Bounds are given in terms of \(E(\sigma^ 2\vee \sigma^{-2}-1)^ k\), where \(\sigma^ 2\vee \sigma^{-2}\) denotes the maximum of the two quantities \(\sigma^ 2\) and \(\sigma^{-2}\), and k is a positive integer, and of \(E(\sigma^ 2-1)^ k\), if k is even.