Fast radio bursts (FRBs) have emerged as powerful probes in cosmology. An optimized method was recently proposed to extract the cosmic baryon density from localized FRBs by maximizing the joint likelihood function of the extragalactic dispersion measure (${\mathrm{DM}}_{\mathrm{ext}}$). In this paper, we identify a crucial factor that was omitted in the probability density function (PDF) for ${\mathrm{DM}}_{\mathrm{ext}}$ in that method. Using simulated FRB data, we demonstrate that neglecting this factor leads to a systematic bias in the inferred cosmic baryon density, with deviations exceeding the $1\sigma$ confidence level. This highlights the necessity of including the missing factor for reliable cosmological applications of FRBs. Furthermore, applying our corrected PDF to a sample of 88 real localized FRBs, we find that the baryon density inferred with the original PDF is inconsistent with the Planck 2018 CMB results, whereas our corrected PDF yields excellent agreement.

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• Cosmological parameters
• Radio, microwave, & sub-mm astronomy