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In this paper we study the probability distribution of the position of atagged particle in the $q$-deformed Totally Asymmetric Zero Range Process($q$-TAZRP) with site dependent jumping rates. For a finite particle system, itis derived from the transition probability previously obtained by Wang andWaugh. We also provide the probability distribution formula for a taggedparticle in the $q$-TAZRP with the so-called step initial condition in whichinfinitely many particles occupy one single site and all other sites areunoccupied. For the $q$-TAZRP with step initial condition, we provide aFredholm determinant representation for the probability distribution functionof the position of a tagged particle, and moreover we obtain the limitingdistribution function as the time goes to infinity. Our asymptotic result for$q$-TAZRP with step initial condition is comparable to the limitingdistribution function obtained by Tracy and Widom for the $k$-th leftmostparticle in the asymmetric simple exclusion process with step initial condition(Theorem 2 in Commun. Math. Phys. 290, 129--154 (2009)).

Distributions of a particle's position and their asymptotics in the  $q$-deformed totally asymmetric zero range process with site dependent  jumping rates