  This paper considers estimation of a random variable in Poisson noise with
signal scaling coefficient and dark current as explicit parameters of the noise
model. Specifically, the paper focuses on properties of the conditional mean
estimator as a function of the scaling coefficient, the dark current parameter,
the distribution of the input random variable and channel realizations. With
respect to the scaling coefficient and the dark current, several identities in
terms of derivatives are established. For example, it is shown that the
gradient of the conditional mean estimator with respect to the scaling
coefficient and dark current parameter is proportional to the conditional
variance. Moreover, a score function is proposed and a Tweedie-like formula for
the conditional expectation is recovered. With respect to the distribution,
several regularity conditions are shown. For instance, it is shown that the
conditional mean estimator uniquely determines the input distribution.
Moreover, it is shown that if the conditional expectation is close to a linear
function in terms of mean squared error, then the input distribution is
approximately gamma in the L\'evy distance. Furthermore, sufficient and
necessary conditions for linearity are found. Interestingly, it is shown that
the conditional mean estimator cannot be linear when the dark current parameter
of the Poisson noise is non-zero.
