  The redshift distribution of fast radio bursts (FRBs) is not well
constrained. The association of the Galactic FRB 200428 with the young magnetar
SGR 1935+2154 raises the working hypothesis that FRB sources track the star
formation history of the universe. The discovery of FRB 20200120E in
association with a globular cluster in the nearby galaxy M81, however, casts
doubts on such an assumption. We apply the Monte Carlo method developed in a
previous work to test different FRB redshift distribution models against the
recently released first CHIME FRB catalog in terms of their distributions in
specific fluence, external dispersion measure ($\rm DM_E$), and inferred
isotropic energy. Our results clearly rule out the hypothesis that all FRBs
track the star formation history of the universe. The hypothesis that all FRBs
track the accumulated stars throughout history describes the data better but
still cannot meet both the $\rm DM_E$ and the energy criteria. The data seem to
be better modeled with either a redshift distribution model invoking a
significant delay with respect to star formation or a hybrid model invoking
both a dominant delayed population and a subdominant star formation population.
We discuss the implications of this finding for FRB source models.
