Noncentral Student,s t-Distribution
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10-4-2021
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Noncentral Student's t-Distribution
A generalization of Student's 
-distribution known as the noncentral Student's 
-distribution is given by
![P(x)=(n^(n/2)n!)/(2^ne^(lambda^2/2)(n+x^2)^(n/2)Gamma(1/2n))<span style=]() {(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))}, " src="https://mathworld.wolfram.com/images/equations/NoncentralStudentst-Distribution/NumberedEquation1.gif" style="height:129px; width:390px" /> |
(1)
|
where 
is the gamma function and 
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
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