(5) In a study of a lake\'s fish population, scientists capture fish from the la

Question

(5) In a study of a lake's fish population, scientists capture fish from the lake, then tag and release them. Suppose that over a period of five days, 200 fish of a certain type are tagged and released. As part of the same study, 20 such fish are captured three days later. Let X denote the number of tagged fish among the 20 captured. Suppose it is known that the lake has 1000 fish of this particular type. (a) What is this distribution? What are its parameter(s)? (b) What is the probability that at most four tagged fish wi be found in the sample? (c) On average, how many tagged fish would you expect to find? Also calculate VX and SDX

Explanation / Answer

Quesetion 5

(a) Here the distribution is HyperGeometric and its parameters are

N = 1000, K = 200 , n = 20 and we have to find the probability of x, which is the number of tagged fish among the 20 captured.

P(X) = 200CX800C(20-X) /1000C20

(b) P(X < = 4) = P(0) + P(1) + P(2) + P(3) + P(4)

=  200C0800C20/1000C20 +  200C1800C19/1000C20 +  200C2800C18/1000C20 +  200C3800C17/1000C20 +  200C4800C16/1000C20

= 0.6300

(d) Here E[X] = 20 * 200/1000 = 4

Var[X] = n (K/N) (N-K)/N (N-n)/(n-1)

= 20 * (200/1000) * (1000 - 200)/1000 * (1000 - 20)/ (1000 - 1)

= 3.1391391

SD[X] = sqrt(3.1391391) = 1.772

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