  The one-bit quanta image sensor is a photon-counting device that produces
binary measurements where each bit represents the presence or absence of a
photon. In the presence of read noise, the sensor quantizes the analog voltage
into the binary bits using a threshold value $q$. The average number of ones in
the bitstream is known as the bit-density and is often the sufficient
statistics for signal estimation. An intriguing phenomenon is observed when the
quanta exposure is at the unity and the threshold is $q = 0.5$. The bit-density
demonstrates a complete insensitivity as long as the read noise level does not
exceeds a certain limit. In other words, the bit density stays at a constant
independent of the amount of read noise. This paper provides a mathematical
explanation of the phenomenon by deriving conditions under which the phenomenon
happens. It was found that the insensitivity holds when some forms of the
symmetry of the underlying Poisson-Gaussian distribution holds.
