Here sgn(nx − k) denotes the sign function: More generally, the mean of n independent uniformly distributed random variables on the interval [a,b] would have the probability density function (PDF) of With a few modifications, the Bates distribution encompasses the uniform, the triangular, and, taking the limit as n goes to infinity, also the normal Gaussian distribution.

will create a similar distribution with a constant variance, such as unity.

A Bates distribution that has been generalized as previously stated fulfills the same purpose for short tail data.

The Bates distribution has an application to beamforming and pattern synthesis in the field of electrical engineering.

The distribution was found to increase the beamwidth of the main lobe, representing an increase in the signal of the radiation pattern in a single direction, while simultaneously reducing the usually undesirable[3] sidelobe levels.