MRI phase contrast is a well known technique for computing the average velocity associated to each pixel. In this work, we calculate the exact probability distribution function for the velocity given the noise in the signal. This pdf is not necessarily Gaussian, particularly for low Signal-to-Noise ratio. We first find the pdf of the signals phase, assuming Gaussian noise in the real and imaginary channels of the signal. The pdf of the velocity is then the convolution of the phases pdfs. To confirm this, we measure several times the velocity in a flow phantom and compare the empirical histogram with the theoretical pdf. We also acquire the velocity from a volunteers aorta using a standard protocol for 4D Flow and multiple coils. Based on this noise characterization, we also propose an optimal weighing for combining multiple coils which is not based only on the coil sensitivities.