We implement an electron avalanche photodiode (e-APD) in the MIRC-X instrument, upgrade of the 6-telescope near-infrared imager MIRC, at the CHARA array. This technology should improve the sensitivity of near-infrared interferometry. We first used the classical Mean-Variance analysis to measure the system gain and the amplification gain. We then developed a physical model of the statistical distribution of the camera output signal. This model is based on multiple convolutions of the Poisson statistic, the intrinsic avalanche gain distribution, and the observed distribution of the background signal. At low flux level, this model constraints independently the incident illumination level, the total gain, and the excess noise factor of the amplification. We measure a total transmission of $48pm3%$ including the cold filter and the Quantum Efficiency. We measure a system gain of 0.49 ADU/e, a readout noise of $10$ ADU, and amplification gains as high as 200. These results are consistent between the two methods and therefore validate our modeling approach. The measured excess noise factor based on the modeling is $1.47pm0.03$, with no obvious dependency with flux level or amplification gain. The presented model allows measuring the characteristics of the e-APD array at low flux level independently of preexisting calibration. With $<0.3$,electron equivalent readout noise at kilohertz frame rates, we confirm the revolutionary performances of the camera with respect to the PICNIC or HAWAII technologies. However, the measured excess noise factor is significantly higher than the one claimed in the literature ($<$1.25), and explains why counting multiple photons remains challenging with this camera.
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