If Y1 has a standard normal distribution, if Y2 is independent of Y1 and has a chi-squared distribution with ν degrees of freedom, and if λ is a non-zero constant, then X, given by is said to have a non-central t-distribution with ν degrees of freedom and non-centrality parameter λ. The probability density function f of X is given by where Γ is the gamma function. The mean and variance of X are respectively (with ν>2). For large values of ν the distribution of X is approximately a normal distribution with mean λ and variance 1.
Subjects: Probability and Statistics.