Variance of hatched eggs from a Poisson laying count

Question

A hen lays a Poisson-distributed number of eggs per month with mean $6$ . Each egg hatches with probability $0.3$ , independently of every other egg and of the number laid. What is the variance of the number of eggs that hatch in a month?

Short answer

$\tfrac95$ . Independently thinning a Poisson count leaves a Poisson count, here with mean $6\times0.3=1.8$ , and a Poisson variable's variance equals its mean. (The law of total variance gives the same: $6\cdot0.21+0.09\cdot6=1.8$ .)

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Lessons behind this question

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

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