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Noncentral Student's t-Distribution

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A generalization of Student's t-distribution known as the noncentral Student's t-distribution is given by

P(x)=(n^(n/2)n!)/(2^ne^(lambda^2/2)(n+x^2)^(n/2)Gamma(1/2n)){(sqrt(2)lambdax_1F_1(1/2n+1;3/2;(lambda^2x^2)/(2(n+x^2))))/((n+x^2)Gamma[1/2(n+1)])+(_1F_1(1/2(n+1);1/2;(lambda^2x^2)/(2(n+x^2))))/(sqrt(n+x^2)Gamma(1/2n+1))},
(1)

where Gamma(z) is the gamma function and _1F_1(a;b;z) is a confluent hypergeometric function of the first kind.

This distribution is implemented in the Wolfram Language as NoncentralStudentTDistribution[n, lambda].

The mean and variance are given by

mu=lambdasqrt(n/2)(Gamma(1/2(n-1)))/(Gamma(1/2n))
(2)
sigma^2=((lambda^2+1)n)/(n-2)-(lambda^2n[Gamma(1/2(n-1))]^2)/(2[Gamma(1/2n)]^2).
(3)

See also

Student's t-Distribution

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Cite this as:

Weisstein, Eric W. "Noncentral Student's t-Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NoncentralStudentst-Distribution.html

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