64. (i) If the joint distribution of X and Y is the bivariate normal distribution (70), then the conditional distribution of Y given X is the normal distribution with variance T2 (1 - p2) and mean 'II -I- (pT/a)(x - n (ii) Let (XI' YI ) , . . . , (X", y,,) be a sample from a bivariate normal distribution, let R be the sample correlation coefficient, and sU'ppose that p = o. Then the conditional distribution of..;t1'=2R/~2 given Xl • • • .• X" is Student's r-distribution with n - 2 degrees of freedom provided L( Xi - X)2 > O. This is therefore also the unconditional distribution of this statistic. (iii) The probability density of R itself is then (84) 1 r[Hn - 1)] (1 _ r2):" -2

[(ii): If Vi = (Xi - X)/V'£(Xj - X)2 SO that '£vi = 0, '£v; = 1, the statistic can be written as LViY; J[L y;2 - ny 2- (I:v;Y;)2]/(n - 2) Since its distribution depends only on p one can assume '1/ = 0, T = 1. The desired result follows from Problem 6 by making an orthogonal transformation from (YI, . . . , Y,,) to (ZI' ···' Zn) such that Zl = {;Y, Z2 = '£v;Y;.]