The redshift distribution of Einstein Probe transients supports their relation to gamma-ray bursts
The launch of the \textit{Einstein Probe} unleashed a new era of high-energy transient discovery in the largely unexplored soft X-ray band. The \textit{Einstein Probe} has detected a significant number of fast X-ray transients that display no gamma-ray emission, complicating their robust association to more common gamma-ray bursts. To explore their possible connection, we analyzed the redshift distribution of both \textit{Einstein Probe} fast X-ray transients and long duration gamma-ray bursts. A comparative analysis of their cumulative redshift distributions using non-parametric two-sample tests, namely the Kolmogorov-Smirnov and Anderson-Darling tests, finds no statistically significant difference. These tests favor that their redshifts are drawn from the same underlying distribution. This empirical connection between \textit{Einstein Probe} transients and long gamma-ray bursts is further supported by their agreement with the so-called ``Amati relation'' between the spectral peak energy and the isotropic-equivalent energy. Together, these results indicate that most extragalactic \textit{Einstein Probe} fast X-ray transients are closely related to long gamma-ray bursts and originate from a massive star (collapsar) progenitor channel. Our findings highlight the role of the \textit{Einstein Probe} in uncovering the missing population of failed jets and dirty fireballs that emit primarily at soft X-ray wavelengths.