2. Noncentral X2-distribution· . (i) If X is distributed as N(!/t, 1), the probability density of V = X2 is pf( v) =. f.f'-OPk(!/t )fu+ 1(v), where Pk(!/t) = (!/t2/2)ke-(lj2),y2/k! and where fu + 1 is the probability density ofax2-variable with 2k + 1 degrees of freedom. (ii) Let Y•• . . . , y,. be independently normally distributed with unit variance and means lI•• .. . , lI, . Then U = f.Y? is distributed according to the noncentral X2 -distribution with r degrees of freedom and noncentrality parameter !/t2 =

f. ~_.lI;. which has probability density (107) 00 pf(u) = L Pk(!/t)f,·+u(u) . k -O Here Pd!/t) and j,+u (u) have the same meaning as in (i), so that the distribution is a mixture of X2-distributions with Poisson weights. [(i): This is seen from

e- ~(,y2+1 ')( e,y.,t> + .- .,t» pf( v) = 2/27TV "The literature on noncentral X2• including tables. is reviewed in Chapter 28 of Johnson and Kotz (1970. Vol. 2). in Chou. Arthur. Rosenstein, and Owen (1984). and in Tiku (1985a).

by expanding the expression in parentheses into a power series, and using the fact that f(2k) = 2u -'f(k)f(k + t)/f;. (ii): Consider an orthogonal transformation to Z" .. . , Z, such that Z, = E'IJ;Y;/o/. Then the Z's are independent normal with unit variance and means £(Z,) = 0/ and £(Z;) = 0 for i > 1.]