The StatsNCTPDF function returns the probability distribution function of the noncentral Student-T distribution. df is the degrees of freedom (positive integer) and d is the noncentrality measure.
f ( x ; n , δ ) = n n / 2 n ! 2 n e δ 2 / 2 ( n + x 2 ) n / 2 Γ ( n 2 ) { 2 δ x 1 F 1 ( n 2 + 1 ; 3 2 ; δ 2 x 2 2 ( n + x 2 ) ) ( n + x 2 ) Γ ( n + 1 2 ) + 1 F 1 ( n + 1 2 ; 1 2 ; δ 2 x 2 2 ( n + x 2 ) ) ( n + x 2 ) Γ ( n 2 + 1 ) } \displaystyle f(x ; n, \delta)=\frac{n^{n / 2} n!}{2^{n} e^{\delta^{2} / 2}\left(n+x^{2}\right)^{n / 2} \Gamma\left(\frac{n}{2}\right)}\left\{\frac{\sqrt{2} \delta x_{1} F_{1}\left(\frac{n}{2}+1 ; \frac{3}{2} ; \frac{\delta^{2} x^{2}}{2\left(n+x^{2}\right)}\right)}{\left(n+x^{2}\right) \Gamma\left(\frac{n+1}{2}\right)}+\frac{{ }_{1} F_{1}\left(\frac{n+1}{2} ; \frac{1}{2} ; \frac{\delta^{2} x^{2}}{2\left(n+x^{2}\right)}\right)}{\sqrt{\left(n+x^{2}\right)} \Gamma\left(\frac{n}{2}+1\right)}\right\} f ( x ; n , δ ) = 2 n e δ 2 /2 ( n + x 2 ) n /2 Γ ( 2 n ) n n /2 n ! ⎩ ⎨ ⎧ ( n + x 2 ) Γ ( 2 n + 1 ) 2 δ x 1 F 1 ( 2 n + 1 ; 2 3 ; 2 ( n + x 2 ) δ 2 x 2 ) + ( n + x 2 ) Γ ( 2 n + 1 ) 1 F 1 ( 2 n + 1 ; 2 1 ; 2 ( n + x 2 ) δ 2 x 2 ) ⎭ ⎬ ⎫ References Evans, M., N. Hastings, and B. Peacock, Statistical Distributions , 3rd ed., Wiley, New York, 2000.
See Also Statistical Analysis , StatsStudentCDF , StatsStudentPDF , StatsNCTCDF