Expected hatchlings from Poisson egg laying

Question

A hen lays eggs over a month according to a Poisson process, so the monthly count is Poisson with mean $6$ . Each egg hatches with probability $0.3$ , independently of every other egg and of how many were laid. What is the expected number of eggs that hatch?

Short answer

$\tfrac95$ . Conditional on $X$ eggs, the expected number of hatchlings is $0.3X$ ; the tower property gives $0.3\,\mathbb{E}[X]=0.3\cdot6=1.8$ . (In fact the hatched count is Poisson with mean $1.8$ by thinning.)

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Lessons behind this question

Defining the Poisson Process: Three Equivalent Constructions
Independence
Expectation, Mean, and Variance

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