One way of estimating the number of fish in a lake is the...

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 $\hat{N}$.

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 $\hat{N}$.

Key Concepts

Capture?Recapture Method

This is a statistical technique used to estimate the size of a population when a complete count is impractical. By capturing a subset of the population, marking or tagging them, and then recapturing another sample, one can use the proportion of marked individuals in the second sample to infer the total population size.

Negative Hypergeometric Distribution

This probability distribution models the number of failures (in this case, non?tagged individuals) observed before a fixed number of successes (tagged individuals) is reached in sampling without replacement. It is central to this problem because the random variable of interest, the number of non?tagged fish caught before the rth tagged fish is captured, follows this distribution.

Expectation and Variance in Sampling Distributions

The calculation of the mean (expected value) and variance of a distribution is essential in understanding its central tendency and spread. In problems like this, these moments provide insight into the typical number of non?tagged fish caught and the variability around this number, facilitating estimation of the unknown population parameter.

Method of Moments Estimation

This estimation method involves equating theoretical moments (such as the expectation derived from the distribution) to observed sample moments in order to solve for unknown parameters. Here, it is used to set the expected number of non-tagged fish equal to the observed value, thereby yielding an estimate for the total population size.

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