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Noncentral chi-squared Distribution

The distribution of νi=1(Zi+δi)2 where Zi are independent standard normal variables and δi are constants. λ=νi=1δ2i>0. (In communications it is called the Marcum-Q function). Can be thought of as a Generalized Rayleigh-Rice distribution. For x>0

f(x;ν,λ)=e(λ+x)/212(xλ)(ν2)/4I(ν2)/2(λx)F(x;ν,λ)=j=0{(λ/2)jj!eλ/2}Pr[χ2ν+2jx]G(q;ν,λ)=F1(x;ν,λ)
μ=ν+λμ2=2(ν+2λ)γ1=8(ν+3λ)(ν+2λ)3/2γ2=12(ν+4λ)(ν+2λ)2

Implementation: scipy.stats.ncx2