Noncentral 2-distribution. 11 (i) If X is distributed as N(, 1), the probability density of V =

Noncentral χ2-distribution.

11

(i) If X is distributed as N(ψ, 1), the probability density of V = X2 is P V

ψ (v) =

∞

k−0 Pk(ψ)f2k+1(v), where Pk(ψ)=(ψ2/2)ke−(1/2)ψ2

/k! and where f2k+1 is the probability density of a χ2-variable with 2k + 1 degrees of freedom.

(ii) Let Y1,...,Yr be independently normally distributed with unit variance and means η1,...,ηr. Then U = Y 2 i is distributed according to the noncentral χ2-distribution with r degrees of freedom and noncentrality parameter ψ2 = r i=1 η2 i , which has probability density pU ψ (u) = ∞
k=0 Pk(ψ)fr+2k(u). (7.60)
Here Pk(ψ) and fr+2k(u) have the same meaning as in (i), so that the distribution is a mixture of χ2-distributions with Poisson weights.
[(i): This is seen from pV ψ (v) = e− 1 2 (ψ2+v)
(eψ√v + e−ψ√v)
2 √2πv by expanding the expression in parentheses into a power series, and using the fact that Γ(2k)=22k−1Γ(k)Γ(k + 1 2 )/
√π.

(ii): Consider an orthogonal transformation to Z1,...,Zr such that Z1 = ηiYi/ψ. Then the Z’s are independent normal with unit variance and means E(Z1) = ψ and E(Zi) = 0 for i > 1.]