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)(ν−2)/4I(ν−2)/2(√λx)F(x;ν,λ)=∞∑j=0{(λ/2)jj!e−λ/2}Pr[χ2ν+2j≤x]G(q;ν,λ)=F−1(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