I'm working on this problem:
Problem: A biologist wants to estimate the amount of fish $N$ in a lake: first he catches $k$-amount of fish ($k < N$), tags these fish and then releases them back in the lake. Now he starts to fish until he catches a tagged one (each fish is always released back in the lake). Let $X_1$ be the amount of fish he has to catch until he catches a tagged one (the tagged one included). He repeats this experiment $n$ times and obtains in the same way $X_2, \ldots, X_n$. The probability to catch a tagged fish is thus $p = k/N$.
a) Find an unbiased estimator for $N$ by only using $X_1$ (tip: find the distribution and expectation value of $Y_1 = X_1 - 1$).
b) Find an unbiased estimator for $N$ by using $X_1, \ldots, X_n$.
Attempt at solution:
a) I think $Y_1$ is geometrically distributed, that is $Y_1 \sim Geom(p). $ So we know that $P(Y_1 = j) = (1-p)^j p$. The expectation value of $Y_1$ is then $$E(Y_1) = \frac{1 - p}{p} = \frac{1 - k/N}{k/N} = \frac{N}{k} - 1. $$ Since I want this to be an unbiased estimator, we need $E(Y_1 ) = N$, that is $$ \frac{N}{k} - 1 = N $$ or $$ k = \frac{N}{N+1}. $$
b) For this part, I don't know what distribution to use when I take into account $X_1, \ldots, X_n$.
Can someone also tell me if part a) is correct, and provide some help with part b)? Thank you.
You already have $$\frac{N}{k}-1 = E(Y_1)=E(X_1-1) = E(X_1)-1.$$ Solve this for $N$, and you get $$N = k\cdot E(X_1) = E(kX_1).$$ Thus, $\hat{N}:=k\cdot X_1$ is an unbiased estimator for $N$.
In b), you can use any $X_i$ to construct an unbiased estimator for $N$ in the same way: $k\cdot X_i$ is an unbiased estimator. Then, the average of all these unbiased estimators, $k\cdot\bar{X}$, is also unbiased: $$E(k\cdot\bar{X}) = E(k\cdot\frac{1}{n}\sum_{i=1}^nX_i)=\frac{1}{n}\sum_{i=1}^n E(k\cdot X_i) =\frac{1}{n}\sum_{i=1}^n N = N.$$
As a matter of fact, you can construct an unbiased estimator from any convex sum of the $X_i$: $$E(k\cdot \sum_{i=1}^n a_i X_i)=N$$ for $\sum{a_i}=1.$
(Follow-up question: how should you choose the $a_i$ to minimize the variance of your estimator?)