Chapter 5.9, Problem 121E

Refer to Exercise 5.117. Suppose that the number N of alligators in the population is very large, with p₁ = .3 and p₂ = .1.

a Find the probability that, in a sample of five alligators, Y₁ = 2 and Y₂ = 1.

b If n = 5, find $E\left(\frac{Y_1}{n}-\frac{Y_2}{n}\right)$ and $V\left(\frac{Y_1}{n}-\frac{Y_2}{n}\right)$.

5.117 A population of N alligators is to be sampled in order to obtain an approximate measure of the difference between the proportions of sexually mature males and sexually mature females. Obviously, this parameter has important implications for the future of the population. Assume that n animals are to be sampled without replacement. Let Y₁ denote the number of mature females and Y₂ the number of mature males in the sample. If the population contains proportions p₁ and p₂ of mature females and males, respectively (with p₁ + p₂ < 1), find expressions for

$E\left(\frac{Y_1}{n}-\frac{Y_2}{n}\right)$ and $V\left(\frac{Y_1}{n}-\frac{Y_2}{n}\right).$