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We study the effects of the beaming-induced luminosity function on statisticsof observed GRBs, assuming the cosmological scenario. We select and divide theBATSE 4B data into 588 long bursts (T$_{90}>2.5$ sec) and 149 short bursts(T$_{90}<2.5$ sec), and compare the statistics calculated in each subgroup. The$<V/V_{\rm max}>$ of the long bursts is $ 0.2901\pm 0.0113$, and that of theshort bursts is $0.4178\pm 0.0239$, which is a Euclidean value. For luminosityfunction models, we consider a cylindrical-beam and a conic-beam. We take intoaccount the spatial distribution of GRB sources as well. A broad luminosityfunction is naturally produced when one introduces beaming of GRBs. Wecalculate the maximum detectable redshift of GRBs, $z_{\rm max}$. The estimated$z_{\rm max}$ for the cylindrical-beam case is as high as $\sim 14$ for thelong bursts and $\sim 3$ for the short bursts. The large $z_{\rm max}$ valuefor the short bursts is rather surprising in that the $<V/V_{\rm max}>$ forthis subgroup is close to the so-called Euclidean value, 0.5. We calculate thefraction of bursts whose redshifts are larger than a certain redshift $z'$,i.e. $f_{\rm > z'}$. When we take $z'=3.42$ and apply the luminosity functionderived for the cylindrical-beam, the expected $f_{\rm > z'}$ is $\sim 75 %$for long bursts. When we increase the opening angle of the conic beam to$\Delta \theta =3^\circ.0$, $f_{\rm > z'}$ decreases to $\sim 20 %$ at $ {\rmz'=3.42}$. We conclude that the beaming-induced luminosity functions arecompatible with the redshift distribution of observed GRBs and that theapparent Euclidean value of $<V/V_{\rm max}>$ may not be due to the Euclideanspace distribution but to the luminosity distribution.Comment: Accepted for publication in the Astronomical Journal (vol. 548, Feb.  20 2001

Effects of Luminosity Functions Induced by Relativistic Beaming on the Statistics of Cosmological Gamma‐Ray Bursts