Problem $5.$ Armand is observing a nuclear decay process which he models as a Poisson process. His supervisor told him to expect $1$ photon every $2$ hours.

Conditional on him waiting at least $5$ hours, what is the probability that he has to wait $8.$

Let $x$ be the number of photons Armand observes in the first $4$ hours. Let $Y$ be the number of photons that Armand sees in hours $2-6.$ Compute the expected value of $xY$