We now give two further estimators that can be found by the method of moments.

Question: Sex ratio of Siberian tigers

The probability of a newborn tiger being male is an unknown parameter, \(\pi\). Assuming that the sexes of tigers in a litter are independently determined, the number of males in a litter of size three will be

\[ X \;\; \sim \; \; \BinomDistn(3, \pi) \]

A researcher recorded the numbers of males from a sample of \(n = 207\) such litters, as summarised by the following frequency table.

Number of males 0 1 2 3
Frequency 33 66 80 28

What is the method of moments estimate of \(\pi\)?

(Solved in full version)

This is an unbiased estimator, but the method of moments estimators sometimes results in an estimator whose bias is non-zero.

Question: Sample from a geometric distribution

If \(\{X_1, X_2, \dots, X_n\}\) is a random sample from a geometric distribution with probability function

\[ p(x) = \pi (1-\pi)^{x-1} \quad \quad \text{for } x = 1, 2, \dots \]

what is the method of moments estimator of \(\pi\)? Is it unbiased?

(Solved in full version)