One dimensional quaternion linear canonical transform in probability theory

The quaternion linear canonical transform is considered as central piece in different fields like applied mathematics, engineering, computer sciences, and statistics. In this work, we have introduced the one-dimensional quaternion linear canonical transform, which is a generalization of the quaternion Fourier transform. We explore the characteristic function within the framework of quaternion algebra, elucidating its essential properties. Building on this foundation, we derive the expected value and variance, offering deeper insights into the probabilistic structures associated with the quaternions.