We derive simple analytic expressions for the flux and spectrum of ultra-high
energy cosmic-rays (UHECRs) predicted in models where the CRs are protons
produced by extra-Galactic sources. For a power-law scaling of the CR
production rate with redshift and energy, d\dot{n} /dE\propto E^-\alpha
(1+z)^m, our results are accurate at high energy, E>10^18.7 eV, to better than
15%, providing a simple and straightforward method for inferring d\dot{n}/dE
from the observed flux at E. We show that current measurements of the UHECR
spectrum, including the latest Auger data, imply
E^2d\dot{n}/dE(z=0)=(0.45\pm0.15)(\alpha-1) 10^44 erg Mpc^-3 yr^-1 at E<10^19.5
eV with \alpha roughly confined to 2\lesseq\alpha<2.7. The uncertainty is
dominated by the systematic and statistic errors in the experimental
determination of individual CR event energy, (\Delta E/E)_{sys} (\Delta
E/E)_{stat} ~20%. At lower energy, d\dot{n}/dE is uncertain due to the unknown
Galactic contribution. Simple models in which \alpha\simeq 2 and the transition
from Galactic to extra-Galactic sources takes place at the "ankle", E ~10^19
eV, are consistent with the data. Models in which the transition occurs at
lower energies require a high degree of fine tuning and a steep spectrum,
\alpha\simeq 2.7, which is disfavored by the data. We point out that in the
absence of accurate composition measurements, the (all particle) energy
spectrum alone cannot be used to infer the detailed spectral shapes of the
Galactic and extra-Galactic contributions.Comment: 9 pages, 11 figures, minor revision