Assume there is an underlying process, $K$, governed by a Poisson distribution, $K\sim Poisson(\lambda)$. We observe realizations this process, but with an imperfect count detection device with failure rate, $p$. I.e. for $k$ counts actually produced by the underlying process, we observe $\tilde{K}=\tilde{k}$ counts, where $\tilde{K}\sim Binomial(k,1-p)$.
What is observed distribution of counts, $P(\tilde{K}=\tilde{k})$?
$P(\tilde{K}=\tilde{k})=\sum_{k=0}^{\infty}P(\tilde{K}=\tilde{k}|k)P(K=k)$
So far I get a diverging sum. This sum should converge, no? Note that we should specially treat the $k=0$ since in this case $P(\tilde{K}=\tilde{k}|k=0)=\delta_{0,\tilde{k}}$.
Any help appreciated.
It's a Poisson distribution with parameter $\lambda(1-p)$. No need to perform the summation; you're just reducing the rate by a factor of $1-p$.