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 PV

ψ (v) = ∞

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



2 = 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 (2k) = 22k−1(k)(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.]