An Exact Determination of the Probability Density Function Under Coordinate Transformations

In this note, the formalism to determine the probability density function (PDF) resulting from a coordinate transformation applied to an arbitrary PDF is developed. The results are applied to the coordinate transformation used in target tracking, specifically the transformation from spherical to Cartesian coordinates. This result is then applied to the specific example of a Gaussian random variable transformed by sine and cosine coordinate transformations. A brief discussion is then made of multi-dimensional transformations.