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PdfNoncentralStudentsT Method
Evaluates the noncentral Student's t probability density function.

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0
Syntax
Copy
public static double NoncentralStudentsT(
	double t,
	double df,
	double delta
)

Parameters

t
Type: SystemDouble
A double value representing the argument at which the function is to be evaluated.
df
Type: SystemDouble
A double value representing the number of degrees of freedom. df must be positive.
delta
Type: SystemDouble
A double value representing the noncentrality parameter.

Return Value

Type: Double
A double value representing the probability density associated with a noncentral Student's t random variable with value t.
Remarks

The noncentral Student's t-distribution is a generalization of the Student's t-distribution. If w is a normally distributed random variable with unit variance and mean \delta and u is a chi-square random variable with \nu degrees of freedom that is statistically independent of w, then

T \;\; = \;\; w/\sqrt{u/\nu}

is a noncentral t-distributed random variable with \nu degrees of freedom and noncentrality parameter \delta, that is, with \nu = df, and \delta = delta. The probability density function for the noncentral t-distribution is:

f(t,\nu,\delta) \;\; = \;\; \frac{\nu^{\nu/2} \; e^{-\delta^2/2}}{\sqrt{\pi} \; \Gamma(\nu/2) \; ( \nu + t^2 ) ^ {(\nu + 1)/2}} \; \sum_{i = 0}^\infty {\Phi_i}

where

\Phi_i \;\; = \;\; \frac{\Gamma((\nu + i + 1)/2)}{i!} \; [\delta t]^i \; \left(\frac{2}{\nu + t^2}\right)^{i/2}

and t = t.

For noncentrality parameter \delta = 0, the PDF reduces to the (central) Student's t PDF:

f(t,\nu,0) \;\; = \;\; \frac{\Gamma((\nu+1)/2) \; \left( 1 \; + \; (t^2/\nu) \right)^{-(\nu+1)/2}}{\sqrt{\nu \pi} \; \Gamma(\nu/2)}

and, for t = 0, the Pdf becomes:

f(0,\nu,\delta) \;\; = \;\; \frac{\Gamma((\nu+1)/2) \; e^{-\delta^2/2}}{\sqrt{\nu \pi} \; \Gamma(\nu/2)}

See Also