67. If (XI' Yd, ...,(Xn , y,,) is a sample from a bivariate normal distribution, the probability density of the sample correlation coefficient R is* (86) 2n - 3 pp(r) = (1 _l)~(n-1)(1 _ r2)~(n -4) '/T(n - 3)! k X L r 2 [Hn+ k - 1)] (2pr) k-O k! or alternatively (87) n - 2 I pp(r) = -'/T-(1 - p2)'(n-l)(1 - r2) ~( -4) 1 /n-2 1 X(J I ~d/ o (1 - pr/)"- 1 - /2 Another form is obtained by making the transformation / = (1 - v)/(1 - pro) in the integral on the right-hand side of (87). The integral then becomes (88) 1 1 (1 - v) n-2 (1 _ pr)~(2n-3) 10 .ffV [1 - !v(1 + pr)]-1/2dv. Expanding the last factor in powers of v, the density becomes (89) where (90) __ n-2r(n-l) I (1 2)2(n -l) I If; r(n _ n - p (1 - r2)2(n-4)(I_ pr)-n+ t XF(l.l. 11+ p 2 r) t2,n- 2;--2 ' eo r(a + j) r(b + j) r

(c) xi F(a,b,c ,x) = L r

(a) r

(b) f(c+j)j! J-O is a hypergeometric function . [To obtain the first expression make a transformation from (Sf, sf, S12) with density (85) to (Sf, sf, R) and expand the factor exp{ pS12/(1 - p2)or ] = "The distribution of R is reviewed by Johnson and K01z (1970, Vol. 2, Section 32) and Patel and Read (1982).

exp{prsls2/(l - p2)O'T} into a power series. The resulting series can be integrated term by term with respect to sf and si. The equivalence with the second expression is seen by expanding the factor (1 - prt)-(n-l) under the integral in (87) and integrating term by term.]