The dynamics of a free electron gas interacting with neutral and excited atoms whose populations are assumed fixed is considered in the case of a three-dimensional model. An analytical form of the time-dependent electron distribution function for arbitrary initial conditions is obtained when interactions are only inelastic and superelastic between two atomic levels. It is shown that this electron gas reaches a stationary state. We deduce an experimental method for the measurement of the population of excited atoms in a neutral gas by injecting a dilute electron beam into the gas.