Two ways to compute probabilities of recognition of shift direction of point on a plane on a background of random rotations

The complex motion of a point on a plane is considered. The observed point and center of the point rotation carry out a parallel shift with probability $p$ in one of $m$ equidistant on an angle of directions at each discrete time moment and, simultaneously, the observed point rotates relatively to this center on a random angle. The decision rule for determining the direction of shift is justified. Expressions for conditional probability densities distribution of sample means of coordinates of the observed point are derived. Formulas for the probabilities of correct recognition of the shift direction are derived in two ways. One way uses the resulting conditional probability density functions. The second way is realized by averaging over random parameters of motion.