Summary: Suppose that \(K \subset \mathbb{C}\) is compact and that \(z_0 \in \mathbb{C}\setminus K\) is an external point. An optimal prediction measure for regression by polynomials of degree at most \(n\), is one for which the variance of the prediction at \(z_0\) is as small as possible. P. G. Hoel and A. Levine [Ann. Math. Stat. 35, 1553–1560 (1964; Zbl 0127.10301)] have considered the case of \(K=[-1,1]\) and \(z_0 = x_0 \in \mathbb{R}\setminus [-1, 1]\), characterizing the optimal measures. More recently, L. Bos et al. [Constr. Approx. 54, No. 3, 431–453 (2021; Zbl 1518.30008)] have given the equivalence of the optimal prediction problem with that of finding polynomials of extremal growth. They also study in detail the case of \(K = [-1, 1]\) and \(z_0 = ia \in i\mathbb{R}\), purely imaginary. In this work we find, for these two cases, the limits of the optimal prediction measures as \(n \rightarrow \infty\) and show that they are the push-forwards via conformal mapping of the Poisson kernel measure for the disk. Moreover, in the case of \(z_0 = ia \in i\mathbb{R}\), we show that the optimal prediction measure of degree \(n\) is actually the Gauss-Lobatto quadrature formula for this limiting push-forward measure.