  Collecting traffic volume data is a vital but costly piece of transportation
engineering and urban planning. In recent years, efforts have been made to
estimate traffic volumes using passively collected probe data that contain
spatiotemporal information. However, the feasibility and underlying principles
of traffic volume estimation based on probe data without pseudonyms have not
been examined thoroughly. In this paper, we present the exact distribution of
the estimated probe traffic volume passing through a road segment based on
probe point data without trajectory reconstruction. The distribution of the
estimated probe traffic volume can exhibit multimodality, without necessarily
being line-symmetric with respect to the actual probe traffic volume. As more
probes are present, the distribution approaches a normal distribution. The
conformity of the distribution was demonstrated through numerical and
microscopic traffic simulations. Theoretically, with a well-calibrated probe
penetration rate, traffic volumes in a road segment can be estimated using
probe point data with high precision even at a low probe penetration rate.
Furthermore, sometimes there is a local optimum cordon length that maximises
estimation precision. The theoretical variance of the estimated probe traffic
volume can address heteroscedasticity in the modelling of traffic volume
estimates.
