Noncentral chi-square distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ noncentral chisquare (δ, n) is used to indicate that the random variable X has the noncentral chi-square distribution with positive integer parameter n and nonnegative noncentrality parameter δ. A noncentral chi-square random variable X with parameters δ and n has probability density function

f(x) = ∑_{k=0}^{∞} \frac{e^{\frac{−δ−x}{2}} ( \frac{δ}{2})^{k}x^{\frac{n+2k}{2} −1}}{( 2^{\frac{n+2k}{2}} ) Γ( \frac{n+2k}{2} )k!} x > 0.

The cumulative distribution, survivor, hazard, cumulative hazard, and inverse distribution on the support of X are mathematically intractable.

The moment generating function of X is

M(t) = E [e^{tX}] = \frac{e^{δt/(1−2t)}}{(1 − 2t)^{n/2}} 2t < 1.

The characteristic function of X is

ϕ(t) = E [e^{itX}] = \frac{e^{δit/(1−2it)}}{(1 − 2it)^{n/2}} 2t < 1.

The population mean, variance, skewness, and kurtosis of X are

E[X] = δ + n  V [X] = 2(n + 2δ)  E [( \frac{X − μ}{σ} ) ^{3}] = \frac{2^{3/2}(n + 3δ)}{(n + 2δ)^{3/2}}

E [( \frac{X − μ}{σ} ) ^{4}] = 3 + \frac{12(n + 4δ)}{(n + 2δ)^{2}} .

APPL verification: The APPL statements

assume(delta ¿= 0); assume(n, posint); X := [[x -¿ sum(exp(-delta / 2) * (delta / 2) ˆ k / factorial(k) * exp(-x / 2) *

x ˆ (n / 2 + k - 1) / (2 ˆ (n / 2 + k) * GAMMA(n / 2 + k)), k = 0 .. infinity)], [0, infinity], [”Continuous”, ”PDF”]];

MGF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X);

verify the moment generating function, population mean, variance, skewness, and kurtosis.