[Richard777 (OP)]

The Compton Effect discovered in 1923 represents the interaction between a photon (x ray) and an electron. The interaction is elastic; the photon is “scattered”, and the electron “recoils”.

The Compton equation relates a ratio of momentum to a scattering angle.

The Compton equation also includes a “probability ratio”, or “distribution ratio” which is related to the Lorentz distribution function. Is this true?

See maths attached.

[CPT ArkAngel]

The Compton equation gives you the change in the wavelength. There is no probability involved. It is a result. If you want to link it to a spatial probability distribution it is up to you but explain why you use one dimension and give more context. From what and where are you going?