3. Noncentral F- and beta-distributionI Let Y" . .. , Y,.; Y.+ I' .. ·' Yn be independently normally distributed with common variance (12 and means £(Y;) = 'IJ;(i = 1, ... , r); £(Y;) = 0 (i = s + 1,.. . , n). (i) The probability density of W = E~_, y;2IL7-s+1y;2 is given by (6). The distribution of the constant multiple (n - s) W/ r of W is the noncentral F-distribution. (ii) The distribution of the statistic B = E~_, y;2/O:~-I y;2 + E7-s+1y;2) is the noncentral beta-distribution, which has probability density (108) 00 L Pk(o/)gtr+k.t(n-s)(b), k-O where (109) ) q-I f(p+q bP- I(l-b) , gp.q

(b) = f(p)f(q) O~b~l is the probability density of the (central) beta-distribution