If Z1, Z2,…, Zν are independent standard normal variables (see normal distribution) and δ1, δ2,…, δν are constants then X, given byis said to have a non-central chi-squared distribution with ν degrees of freedom and non-centrality parameter λ, whereThe probability density function f of X is given bywhere Γ is the gamma function. The distribution, which is unimodal, has mean (ν+λ) and variance 2(ν+2λ). If λ=0 then the distribution becomes the chi-squared distribution with ν degrees of freedom.