Estimate the number of red fishes in the pond

[kudesnik]

hi all!

I have the following problem (in reality I study objects which may be perfectly described by "fishes" as below):

Fishes in the pond: There are N fishes (x of blue color and y of red color) in the pond. N>>10^5, x+y=N, x>>y Each fish has a weight w - positive numerical value. Suppose that the fishing process is a random selection among all the fishes (the probability for catching of blue fish is x/N, for red - y/N).
My tasks:
1) Given minimizing the number of catched fishes estimate the number of red fishes in the pond with predefined accuracy (5-15% would be just ok).
2) Given minimizing the number of catched fishes estimate the total weight of all red fishes.

I would appreciate any reference (I believe it certainly exists since the problem looks like very typical!) to a textbook or a web page in which this or similar problem is considered.

 

[HallsofIvy]

When you say "Given minimizing the number of catched fishes" do you mean that you are given some specific number of red and blue fish caught? without that information, I don't see how you can do this.

Edited after a little more thought- Clearly if you catch r red fish and b blue fish, with r+b= n, then the percentage of red fish in your catch is r/n. Given that the best estimate of the number of red fish in the entire lake is (r/n)N. You are asking for the smallest n such that you can still be sure that the estimate is within, say 15% of the true value.

Now, that I am clearer on what you are asking, I have to admit that I don't know enough statistics to answer that!

 

[tacman]

Hi there,

Thank you for sharing your problem with us. Estimating the number of red fishes in a pond is indeed a common problem in many fields such as ecology and fisheries management.

To estimate the number of red fishes in the pond with a predefined accuracy, we can use a statistical sampling method called capture-recapture. This method involves catching and marking a certain number of fishes, releasing them back into the pond, and then catching a second sample and recording how many of them are marked. By using a mathematical formula, we can estimate the total number of fishes in the pond based on the number of marked fishes in the second sample. This method can also be used to estimate the number of red fishes specifically, as we can mark only the red fishes in the first sample.

To estimate the total weight of all red fishes, we can use a similar method called weight-recapture. This involves catching and weighing a sample of fishes, releasing them back into the pond, and then catching a second sample and weighing them again. By using the same mathematical formula, we can estimate the total weight of all fishes in the pond based on the average weight of the second sample.

I recommend checking out some resources on capture-recapture and weight-recapture methods, as well as consulting with a statistician for a more precise estimation. Good luck with your study!