(i) If the joint distribution of X and Y is the bivariate normal distribution (5.69), then the conditional distribution of Y given x is the normal distribution with variance τ 2(1 − ρ2) and mean η + (ρτ /σ)(x − ξ).

(ii) Let (X1, Y1),..., (Xn, Yn) be a sample from a bivariate normal distribution, let R be the sample correlation coefficient, and suppose that ρ = 0. Then the conditional distribution of √n − 2R/√1 − R2 given x1,...,xn, is Student’s t-distribution with n−2 degrees of freedom provided (xi−x¯)

2 > 0.

This is therefore also the unconditional distribution of this statistic.

(iii) The probability density of R itself is then p(r) = 1

√n

Γ[ 1 2 (n − 1)]

Γ[ 1 2 (n − 2)] (1 − r2

)

1 2 n−2

. (5.83)

[(ii): If vi = (x1 − x¯)/

(xj − x¯)2 so that vi = 0, v2 1 = 1, the statistic can be written as

viYi ,Y 2 i − nY¯ 2 − (

viYi)

2

/(n − 2)

.

Since its distribution depends only on ρ one can assume η = 0, τ = 1. The desired result follows from