t_distribution Derived Type

private pure elemental function td_cdf(this, x) result(rst)

Computes the cumulative distribution function.

The CDF for Student's T-Distribution is given as $$F(t) = \int_{-\infty}^{t} f(u) \,du = 1 - \frac{1}{2} I_{x(t)} \left( \frac{\nu}{2}, \frac{1}{2} \right)$$ where $$x(t) = \frac{\nu}{\nu + t^2}$$ .

Arguments

Return Value real(kind=real64)

The value of the function.

private pure function dist_defined_range(this) result(rst)

Gets the defined range for the distribution.

Arguments

Return Value real(kind=real64), dimension(2)

The defined range of the probability distributions [min, max]. In the event that either min or max are infinite, a value of huge(0.0d0) is returned as opposed to infinite to avoid possible issues with using these values directly.

private pure function td_mean(this) result(rst)

Computes the mean of the distribution.

Arguments

Return Value real(kind=real64)

The mean.

private pure function td_median(this) result(rst)

Computes the median of the distribution.

Arguments

Return Value real(kind=real64)

private pure function td_mode(this) result(rst)

Computes the mode of the distribution.

Arguments

Return Value real(kind=real64)

The mode.

private pure elemental function td_pdf(this, x) result(rst)

Computes the probability density function.

The PDF for Student's T-Distribution is given as $$f(t) = \frac{ \Gamma \left( \frac{\nu + 1}{2} \right) } { \sqrt{\nu \pi} \Gamma \left( \frac{\nu}{2} \right) } \left( 1 + \frac{t^2}{\nu} \right)^{-(\nu + 1) / 2}$$ .

Arguments

Return Value real(kind=real64)

The value of the function.

private subroutine td_recenter(this, x)

Recenters the distribution about the supplied value. This routine has no effect for this distribution as it is always centered about 0.

Arguments

private pure elemental function dist_std_var(this, x) result(rst)

Computes the standardized variable for the distribution.

Arguments

Return Value real(kind=real64)

The result.

private pure function td_variance(this) result(rst)

Computes the variance of the distribution.

Arguments

Return Value real(kind=real64)

The variance.