In this note we present a new special function that behaves like the error function and we provide an approximated accurate closed form for its CDF in terms of both Chèbyshev polynomials of the first kind and the error function. Also we provide its series representation using Padé approximant. We show a convincing numerical evidence about an accuracy of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mn mathvariant="normal">1</mml:mn><mml:msup><mml:mrow><mml:mn mathvariant="normal">0</mml:mn></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn mathvariant="normal">6</mml:mn></mml:mrow></mml:msup></mml:math> for the approximants in the sense of the quadratic mean norm. A similar approach may be applied to other probability distributions, for example, Maxwell–Boltzmann distribution and normal distribution, such that we show its application using both of those distributions.