Let $Y, Y' \sim Poisson \left( \lambda, C \right)$, where $\lambda$ is the mean and $C$ is some upper censoring limit. In this censored Poisson distribution, the probability mass beyond $C$ in an uncensored Poisson distribution is aggregated at $C$. Thus, the probability of $Y = C$ accounts for $P(Y > C)$ in the uncensored case.I'm interested in $E \left[Y - Y'\right]$, i.e. the absolute difference between two iid draws from $Poisson \left( \lambda, C \right)$.
For an uncensored Poisson Distribution, this can be calculated in a closed form by using modified Bessel functions, as $E \left[Y - Y'\right] = 2 \lambda^{-2 \lambda} \{ I_{0} \left( 2 \lambda \right) + I_{1} \left( 2 \lambda \right)\}$ as derived in Katti (1960), Moments of the Absolute Difference and Absolute Deviation of Discrete Distributions.
My question is, is there a closed form expression for $E \left[Y - Y'\right]$ in the censored case? Any insights, references, or approaches would be greatly appreciated.
