Two-user fading multiple access channel (MAC) is investigated, which is corrupted by random fading coefficients and additive Gaussian noise. It is assumed that the channel is block fading, and each transmitter knows only its own channel state to the receiver, but does not know the other transmitter's channel state. The receiver has full knowledge of channel state information (CSI). The performance measure, the expected capacity region over channel statistics, is studied for two scenarios. For the first scenario, in which user 1 has multiple states, and user 2 has one state, most part of the boundary of the expected capacity region is characterized. Interestingly, these rate points are also on the boundary of the capacity region (i.e., the best achievable rate pairs) when the CSI is fully known at both transmitters. Furthermore the expected capacity region is fully characterized for some asymptotic regimes. For the second scenario, in which both users 1 and 2 have two states, a number of achievable regions are studied, and are demonstrated to be close to an outer bound numerically.