y-ray bursts may be from jetlike sources in relativistic bulk motion. In this case, the emisison of y rays is strongly beamed along the direction of the bulk motion. We analyze effects of the relativistic bulk motion and the resulting beamed emission on statistics of the y-ray bursts. PACS number(s1: 98.70.Rz, 98.62.Nx, 98.80.H~ The physical origin of y-ray bursts (GRB's) has remained elusive. Major difficulties are mainly due to lack of positive identifications of GRB's with known astrophysical counterparts and to uncertain distance scales. BATSE's (Burst and Transient Source Experiment's) relatively low detection threshold (F,,, = lo-' erg/s cm2) and homogeneous detection have facilitated meaningful statistical analyses [I]. The isotropy of G R B sky positions [2] poses a serious difficulty for galactic or extended halo models [3]. Cosmological models [4] are subject to a rather tight energy constraint at the burst sources 151. In the cosmological scenario, the deviation of the slope of from -1 (for a homogeneous distribution in the Euclidean space) or ( V/V,,, ) < 0.5 is naturally explained [6] while in the halo scenario some special radial distributions are required [3,7]. The lack of source identifications with counterparts is puzzling in the cosmological scenario since any sources residing in galaxies could have been identified with their hosts.If y-ray emission is isotropic, the luminosity L of a burst from a source at a redshift z,km/s Mpc (for our discussions, h =0.5), and a= 1-2 is the photon index [6]. Isotropic y-ray emission from a typical cosmological burst at z -1 roughly requires a total burst energy -10'' erg or a fluence > lo-' erg/cm2. This large energy with short emission time scales [min (duration, variability time scale)] implies that some physics is required for yray photoris to escape from the emission region at all [ S ] . However, if emission is from a jetlike source in relativistic bulk motion, this severe energy constraint could be and ( V/V,,, ) avoided [5,8]. Since no other models convincingly predict relativistically moving sources, plausible candidates could be rare jets with high Lorentz factors from faint unidentified extragalactic objects [quasistellar objects (QSO's) or active galactic nuclei (AGN)]. If extragalactic objects are indeed involved, identification of GRB's with (x-ray, optical, IR, or radio) counterparts is likely. The relativistic bulk motion motivates a physically meaningful G R B luminosity distribution [9], which is otherwise rather arbitrary and unconstrained. We consider effects of relativistic bulk motion and resulting beamed emission on G R B statistics. We assume that sources emit radiation isotropically in their comoving frames. Our statistical analysis is independent of any specific emission mechanisms whose details will be addressed in a separate paper.We assume that sources are described either by a "standard candle" (a standard luminosity with a spectral index) or by a "standard source" (a standard Lorentz factor, a standard intrinsic luminosity in...