Noncentral chi-square distribution (moments and parity-cased PDF)
NoncentralChiSquareDistribution[ν, λ]
Mean[NoncentralChiSquareDistribution[ν, λ]]
→ λ + νPDF[NoncentralChiSquareDistribution[ν, λ], x]
→ Piecewise[{{(E^((-x - λ)/2)*x^(-1 + ν/2)*Hypergeometric0F1Regularized[ν/2, (λ*x)/4])/2^(ν/2), x > 0}}, 0]CDF[NoncentralChiSquareDistribution[ν, λ], x]
→ Piecewise[{{MarcumQ[ν/2, Sqrt[λ], 0, Sqrt[x]], x > 0}}, 0]