naginterfaces.library.stat.prob_​students_​t

naginterfaces.library.stat.prob_students_t(t, df, tail='L')

prob_students_t returns the lower tail, upper tail or two tail probability for the Student’s $t$-distribution with real degrees of freedom.

For full information please refer to the NAG Library document for g01eb

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01ebf.html

Parameters

tfloat

$t$, the value of the Student’s $t$ variate.

dffloat

$\nu$, the degrees of freedom of the Student’s $t$-distribution.

tailstr, length 1, optional

Indicates which tail the returned probability should represent.

$tail=\text{'U'}$

The upper tail probability is returned, i.e., $P(T\ge t:\nu )$.

$tail=\text{'S'}$

The two tail (significance level) probability is returned, i.e., $P(T\ge \left|t\right|:\nu )+P(T\le -\left|t\right|:\nu )$.

$tail=\text{'C'}$

The two tail (confidence interval) probability is returned, i.e., $P(T\le \left|t\right|:\nu )-P(T\le -\left|t\right|:\nu )$.

$tail=\text{'L'}$

The lower tail probability is returned, i.e., $P(T\le t:\nu )$.

Returns

pfloat

Either the lower tail, upper tail or two tail probability for the Student’s $t$-distribution, depending on the value of $tail$ supplied.

Raises

NagValueError

(errno $1$)

On entry, $tail=⟨value⟩$.

Constraint: $tail=\text{'L'}$, $\text{'U'}$, $\text{'S'}$ or $\text{'C'}$.

(errno $2$)

On entry, $df=⟨value⟩$.

Constraint: $df\ge 1.0$.

Notes

The lower tail probability for the Student’s $t$-distribution with $\nu$ degrees of freedom, $P(T\le t:\nu )$ is defined by:

$$P(T\le t:\nu )=\frac{\Gamma \left((\nu +1)/2\right)}{\sqrt{\pi \nu }\Gamma \left(\nu /2\right)}{\int }_{-\infty }^t{[1+\frac{T^2}{\nu }]}^{-(\nu +1)/2}dT\text{,}\nu \ge 1\text{.}$$

Computationally, there are two situations:

1. when $\nu <20$, a transformation of the beta distribution, $P_{\beta }(B\le \beta :a,b)$ is used

   $$P(T\le t:\nu )=\frac{1}{2}{P}_{\beta }(B\le \frac{\nu }{\nu +t^2}:\nu /2,\frac{1}{2})\text{when}t<0.0$$

   or

   $$P(T\le t:\nu )=\frac{1}{2}+\frac{1}{2}{P}_{\beta }(B\ge \frac{\nu }{\nu +t^2}:\nu /2,\frac{1}{2})\text{when}t>0.0\text{;}$$

2. when $\nu \ge 20$, an asymptotic normalizing expansion of the Cornish–Fisher type is used to evaluate the probability, see Hill (1970).

References

Abramowitz, M and Stegun, I A, 1972, Handbook of Mathematical Functions, (3rd Edition), Dover Publications

Hastings, N A J and Peacock, J B, 1975, Statistical Distributions, Butterworth

Hill, G W, 1970, Student’s $t$ -distribution, Comm. ACM (13(10)), 617–619