A production line creates products at a 60% success rate (40% are defective). The number of object produced in an hour is a RV Y which follows a poisson distribution of parameter 20. A RV X represents the number of defective items produced in an hour. Calculate the probability that X=k knowing that Y=n.
This is a homework assignment. I am looking for someone to point me in the right direction as I have no clue how to start.
Since the rate of defection is $\frac{2}{5}$, then if you produce $m$ items in an hour, the expected number of defect items is $\frac{2}{5}m$. Hence $X|y\sim\mathcal{P}\left(\frac{2}{5}y\right)$, and $\Pr(X=k|Y=n)=\frac{2^k n^k e^{-\frac{2}{5}n}}{5^k k!}$