An analytical derivation of the EMG signal's amplitude probability density function (EMG PDF) is presented and used to study how an EMG signal builds-up, or fills, as the degree of muscle contraction increases. The EMG PDF is found to change from a semi-degenerate distribution to a Laplacian-like distribution and finally to a Gaussian-like distribution.We present a measure, the EMG filling factor, to quantify the degree to which an EMG signal has been built-up. This factor is calculated from the ratio of two non-central moments of the rectified EMG signal. The curve of the EMG filling factor as a function of the mean rectified amplitude shows a progressive and mostly linear increase during early recruitment, and saturation is observed when the EMG signal distribution becomes approximately Gaussian. Having presented the analytical tools used to derive the EMG PDF, we demonstrate the usefulness of the EMG filling factor and curve in studies with both simulated signals and real signals obtained from the tibialis anterior muscle of 10 subjects. Both simulated and real EMG filling curves start within the 0.2 to 0.35 range and rapidly rise towards 0.5 (Laplacian) before stabilizing at around 0.637 (Gaussian). Filling curves for the real signals consistently followed this pattern (100% repeatability within trials in 100% of the subjects). The theory of EMG signal filling derived in this work provides (a) an analytically consistent derivation of the EMG PDF as a function of motor unit potentials and motor unit firing patterns; (b) an explanation of the change in the EMG PDF according to degree of muscle contraction; and (c) a way (the EMG filling factor) to quantify the degree to which an EMG signal has been built-up.