Question

One way of estimating the number of fish in a lake is the following capturerecapture sampling scheme. Suppose there are \(N\) fish in the lake where \(N\) is unknown. A specified number of fish \(T\) are captured, tagged, and released back to the lake. Then at a specified time and for a specified positive integer \(r\), fish are captured until the \(r t h\) tagged fish is caught. The random variable of interest is \(Y\) the number of nontagged fish caught. (a) What is the distribution of \(Y ?\) Identify all parameters. (b) What is \(E(Y)\) and the \(\operatorname{Var}(Y)\) ? (c) The method of moment estimate of \(N\) is to set \(Y\) equal to the expression for \(E(Y)\) and solve this equation for \(N .\) Call the solution \(\hat{N}\). Determine \(\hat{N}\). (d) Determine the mean and variance of \(\hat{N}\).

Short answer

(a) \(Y\) is geometrically distributed with parameter \(p=T/N\). (b) The expectation value \(E(Y)=(N - T) / T\) and the variance \(\operatorname{Var}(Y) = (N - T) / T^2\). (c) The method of moment estimate \(\hat{N}\) is \(\hat{N} = T(Y+1)/r\). (d) The mean of \(\hat{N}\) is \(N\) and its variance is \(N^2((r - 1)/r^3)\).

Step-by-step solution

Identifying the distribution of Y

By examining the problem, we can see that the fish in the lake are assumed to be well mixed, that is, each fish has the same chance of being captured which makes the capturing experiment a Geometric distribution. Therefore, \(Y\) as the number of non-tagged fish follows a geometric distribution with parameter \(p = T/N\). So, the Probability Mass Function (pmf) would be \(p(y) = (1-p)^y * p\), \(y\) = 0, 1, 2, ...

Calculating Expectation and Variance

The expectation value and variance for a geometric random variable, \(Y\) are given by \(E(Y) = (1-p)/p\) and \(\operatorname{Var}(Y) = (1-p)/p^2\) respectively. Substitute \(p = T/N\) into these equations.

Method of moments estimate

The method of moments estimate of \(N\) is obtained by setting the sample mean (which is \(Y\) in this case) equal to the theoretical mean (or Expected Value, \(E(Y)\)). Solving this equation for \(N\) gives \(\hat{N} = T(Y+1)/r\).

Mean and Variance of \(\hat{N}\)

The mean and variance of \(\hat{N}\) can be calculated by substituting \(\hat{N}\) into equations of mean and variance which gives \(E[\hat{N}] = N\) and \(\operatorname{Var}(\hat{N}) = N^2((r - 1)/r^3)\).