Given a Poisson process with parameter $q$, and if $X$ is the number of arrivals during $(0, 1]$, then what is $E(X | X > 0)$?
Could I use the memorylessness of the process to write: $1 + E(X)$?
Thanks everyone.
2026-03-27 23:30:58.1774654258
Conditional expectation of a Poisson Process
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Since $X$ has Poisson distribution with mean $q$, we compute $$ \mathbb E\left[X\mid X>0\right] = \frac{\mathbb E\left[X\mathsf 1_{\left\{ X>0\right\}}\right]}{\mathbb P(X>0)} = \frac{\mathbb E\left[X\right]}{\mathbb P(X>0)} = \frac q{1-e^{-q}}.$$