  Motivated by recent real-time electron counting experiments, we evaluate the
full counting statistics (FCS) for the probability distribution of the electron
number inside a quantum dot which is weakly coupled to source and drain leads.
A non-Gaussian exponential distribution appears when there is no dot state
close to the lead chemical potentials. We propose the measurement of the joint
probability distribution of current and electron number, which reveals
correlations between the two observables. We also show that for increasing
strength of tunneling, the quantum fluctuations qualitatively change the
probability distribution of the electron number. In this paper, we derive the
cumulant generating functions (CGFs) of the joint probability distribution for
several cases. The Keldysh generating functional approach is adopted to obtain
the CGFs for the resonant-level model and for the single-electron transistor in
the intermediate conductance regime. The general form for the CGF of the joint
probability distribution is provided within the Markov approximation in an
extension of the master equation approach [D. A. Bagrets, and Yu. V. Nazarov,
Phys. Rev. B {\bf 67}, 085316 (2003)].
