Owen T Function Calculator

Compute Owen T function T(h,a) by numerical integration, with bivariate normal probability and standard normal CDF and density.

Input

Enter h and a to compute the Owen T function T(h,a) by numerical integration. It is a special function used in bivariate normal probability calculations.

h (first argument)

Any real number, often a standardized value such as 1.

a (second argument, upper limit)

Any real number. A negative a flips the sign since T is odd in a.

Result

T(h, a) with h = 1, a = 0.5

0.04306469

Bivariate normal (1/2)Phi(h) minus T(h,a)

0.37760768

Standard normal CDF Phi(h)

0.84134475

Standard normal density phi(h)

0.24197072

Product Phi(h) times (1 minus Phi(h))

0.13348376

Curve of T(h, a) as a varies

Curve of the integrand g(x)

How it works

The Owen T function is the integral T(h,a)=(1/2pi) times the integral from 0 to a of exp(minus h squared times (1 plus x squared) over 2) divided by (1 plus x squared) dx.
This calculator evaluates the integral with the composite Simpson rule over a finely divided interval from 0 to a to obtain T(h,a).
The T function is even in h, so T(minus h, a)=T(h, a), and odd in a, so T(h, minus a)= minus T(h, a).
When a equals 1, the identity 2 times T(h, 1)=Phi(h) times (1 minus Phi(h)) holds. The displayed product Phi(h)(1 minus Phi(h)) lets you check this.
The bivariate normal probability P(X less than or equal to h, Y less than or equal to 0) for correlation rho, with a equal to rho over the square root of (1 minus rho squared), equals one half Phi(h) minus T(h, a).
The standard normal CDF Phi is computed from a self contained error function erf implementation with good accuracy.

Reviews

Tell us what you think of this calculator.