The number of fish in a certain lake is a Pois($\lambda$) random variable. Worried that there might be no fish at all, a statistician adds one fish to the lake. Let Y be the resulting number of fish (so Y is 1 plus a Pois($\lambda$) random variable). (a) Find E(Y$^2$) (simplify). (b) Find E(1/Y) (in terms of $\lambda$; do not simplify yet). (c) Find a simplified expression for E(1/Y). Hint: k!(k+1) = (k+1)!.
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Then X ~ Pois(A) and Y = X + 1. Show more…
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Lien L.
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 $\hat{N}$.
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The Binomial and Related Distributions
Assume that a lake contains a total of N fish. Suppose we catch M among these fish, mark them in some way, and then put them back in the lake. After a certain amount of time, so that the marked fish have mixed well with the other ones in the lake, we select randomly n fish from the lake. Let X be the number of marked fish in that sample. Find the probability function of X.
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