On applying Möbius transformation to cardioid random variables

Möbius transformation from the unit circle onto itself is applied to a cardioid random variable. The distribution function, cosine and sine moments, conditions for modality and symmetry of the resulting distribution are studied. The paper proposes a bivariate cardioid distribution with symmetric cardioid marginals. The distribution is generated from a circular-circular structural model or a method of trivariate reduction introduced and controlled by a Möbius transformation link between common parts of the two variables. An illustration is given for wind direction data as an application of the bivariate cardioid distribution.