9 A worker for the Department of Fish and Game is assigned t

9 A worker for the Department of Fish and Game is assigned the job of estimating the number of trout in a certain lake of modest size. She proceeds as follows: She catches 100 trout, tags each of them, and puts them back in the lake. One month later, she catches 100 more trout, and notes that 10 of them have tags. (a) Without doing any fancy calculations, give a rough estimate of the number of trout in the lake. (b) Let N be the number of trout in the lake. Find an expression, in terms of N, for the probability that the worker would catch 10 tagged out of the 100 trout that she caught the second time. (c) Find the value of N which maximizes the expression in part (b). This value is called the maximum likelihood estimate for the unknown quantity N. Hint: Consider the ratio of the expressions for successive values N.

Solution

a)

Since there were 100 tagged the first time, the sample size would be 100. In the next attempt again there are 100 trout tagged. This time it is observed that 10 (or 10% of the sample size are already tagged) . If this sample represents the population, then 10% of the fish were tagged the first time. Hence the number of trout in the lake can be assumed to be 1000

b) The probability of catching 10 tagged trout among the hundred would be (100C10)*(100/N)^10*(0.9)^90

c) The value of N that would maximize above expression is 100