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Ecologists want to estimate the total number \(N\) of penguins in a colony (\(N>150\)). They initially catch and mark 150 penguins. Later, they perform \(n\) random sightings with replacement and record the proportion of marked penguins observed.
\(n\) (Number of sightings) 100 1,000 10,000
Proportion of marked penguins 0.312 0.295 0.3001
  1. Define a random variable \(X_i\) associated with the \(i\)-th sighting (\(X_i=1\) if the penguin is marked and \(0\) otherwise). What is the distribution of \(X_i\)? Give its parameter \(p\) in terms of \(N\).
  2. Express the proportion of marked penguins observed after \(n\) sightings as a sample mean \(\overline{X}_n\).
  3. Using the Law of Large Numbers and the result for \(n=10\,000\), give an estimate of \(p\).
  4. Deduce an estimate of the total population \(N\) by solving the equation linking \(p\) and \(N\).

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