On the Distribution of the Conditional Mean Estimator in Gaussian Noise

Consider the conditional mean estimator of the random variable X from the noisy observation Y = X + N where N is zero mean Gaussian with variance σ2 (i.e., $\mathbb{E}\left[ {\left. X \right|Y} \right]$). This work characterizes the probability distribution of $\mathbb{E}\left[ {\left. X \right|Y} \right]$. As part of the proof, several new identities and results are shown. For example, it is shown that the k-th derivative of the conditional expectation is proportional to the (k + 1)-th conditional cumulant. It is also shown that the compositional inverse of the conditional expectation is well-defined and is characterized in terms of a power series by using Lagrange inversion theorem.