Titipounamu (rifleman) are our (NZ's) smallest bird. For this question we are going to consider an area of bush where there are thought to be N 200 titipounamu, of which a random sample of 50 have been captured, tagged and returned to the bush. Tieke (saddleback) and tui also live in this area of bush. = a What does "random sample" mean about the probability of capture 200 of all sets of 50 titipounamu? (1) 50 b Suppose that a new random sample of titipounamu is dra.vn, this time with size 20. This is an important method that is widely used in ecology known as capture-recapture. i Consider the discrete distributions named in Section 2.1 of your course book. Let K be the number of previously tagged titipounamu in this new sample of 20. Name the distribution and its parameter/s that could be used to model K. (You may assume that a titipounamu that was captured and tagged before doesn't become more or less likely to be captured again.) (2) ii What is the probability that exactly 5 of the 20 titipounamu in the new sample were previously tagged? (1) (1) iv Find the variance, Var(K), and standard deviation, SD(K), of K. (2) iii What is the expected number of previously tagged titipounamu in the new sample? Suppose that X ~ Poisson (1) and Y Poisson (2) are independent random variables with Var(X) + Var(Y) = 6. Find P(X + Y < 3). (4)