The probability density function (pdf) of the received signal of an ambient backscatter communication system is derived, assuming that on-off keying (OOK) is performed at the tag, and that the ambient radio frequency (RF) signal is white Gaussian. The pdf of the received signal is then utilized to design two different types of non-coherent detectors. The first detector directly uses the received signal to perform a hypothesis test. The second detector first estimates the channel based on the observed signal and then performs the hypothesis test. Test statistics and optimal decision threshold of the detectors are derived. The energy detector is shown to be an approximation of the second detector. For cases where the reader is able to avoid or cancel the direct interference from the RF source (e.g., through successive interference cancellation), a third detector is given as a special case of the first detector. Numerical results show that both the first and the second detectors have the same bit error rate (BER) performance, making the second detector preferable over the first detector due to its computational simplicity.