Distance probability function determined from parallax and accurate flux with known spatial and luminosity distributions. The black line gives the values based on parallax alone (reproducing the black line in Fig. 3). For a lognormal luminosity function, the dashed blue line shows the extra term D²f_L(L_oD²S′) and the solid blue line the overall distribution according to Eq. (24) for values appropriate for PSR J0218+4232. The dashed and solid red lines and the brown lines idem for power-law luminosity functions both with L_min = 0.1 mjy kpc² and L_max = 10 mJy kpc² for α = −1, and L_max = 100 mJy kpc² for α = −2, respectively. All curves are normalized to their maximum value.