A system of equations is derived in order to predict the first moments of the dependent variables in moist, deep convection. Closure of the non-linear system is achieved by making the covariances of the dependent variables proportional to an eddy diffusivity. The eddy diffusivity is proportional to the turbulent kinetic energy and to a characteristic time scale of the turbulence. Although the time scale is simply an algebraic function of the mean strain rate and the Richardson number, an additional partial differential equation is required to determine the turbulent energy. Cloud water is assumed to form instantaneously when the air becomes saturated. Consequently the average amount of cloud water depends upon the probability distribution of the total water (vapour plus liquid). To estimate the average cloud water, the total water is approximated by a binary switching process. Part II of the work describes a finite element method which is used to solve numerically the system of equations.