The following problem comes from a stats course taught by Joe Blitzstein in Fall ’09–my freshman year of college. Suppose a chicken lays eggs, where is Poisson distributed with parameter . Given that the chicken has layed eggs, each egg hatches independently of the other eggs with probability . Mathematically speaking, if denotes the number of hatched eggs, the random variable is binomial distributed with parameters . What is the distribution of ?

We use conditional probability to tackle this problem. Specifically, we condition on the number of eggs the chicken lays.

The infinite series in the last expression should be recognizable to the reader as the Taylor expansion of . Substituting this in, we obtain

But this last expression is the probability mass function (PMF) of a Poisson random variable with parameter evaluated at .

Consider now the number of eggs which don’t hatch, which is the random variable ? At first glance, it seems that and are highly correlated. When I first encountered this problem, I thought this was indeed the case. However, it turns out that and are independent. To see this, we show that the joint distribution of and factors into the product of the marginal distributions. A completely analogous argument to that above shows that the marginal distribution for is Poisson with parameter . Conditioning on , we have that

Observe that

so that

What if instead of taking the probability that an egg hatches to be a known constant, we treat as a random variable? I know nothing about chicken egg hatching probabilities, so I’m not sure what would be a good choice of distribution for . Any thoughts?

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