  The K$\beta$ transition in muonic hydrogen was measured with a
high-resolution crystal spectrometer. The spectrum is shown to be sensitive to
the ground-state hyperfine splitting, the corresponding triplet-to-singlet
ratio, and the kinetic energy distribution in the $3p$ state. The hyperfine
splitting and triplet-to-singlet ratio are found to be consistent with the
values expected from theoretical and experimental investigations and,
therefore, were fixed accordingly in order to reduce the uncertainties in the
further reconstruction of the kinetic energy distribution. The presence of
high-energetic components was established and quantified in both a
phenomenological, i.e. cascade-model-free fit, and in a direct deconvolution of
the Doppler broadening based on the Bayesian approach.
