Question 1. [50 marks] In a certain region, there is an average of 0.6 ant nests per square kilometre and an average of 0.3 beehives per square kilometre. Suppose that the locations of ant nests are independent of other ant nests, the locations of beehives areindependent of other beehives, and the locations of ant nests and beehives are independent of each other. (a) [5 marks] LetA and B be the counts of ant nests and beehives, respectively, in a randomly chosen square kilometre. Justify why A follows a Poisson distribution with A = 0.6 and 8 follows a Poisson distribution with A = 0.3. (b) [10 marks] Let S be the count of ant nests and beehives in a randomly chosen square kilometre. That is, let S = A + B. Use the moment generating function method to show that 5 follows a Poisson distribution with A = 0.9. Note: A Poisson random variableX with parameterl has moment generating function: MX(t) = exp [Met 1)] (c) [5 marks) Find the probability that in a randomly chosen square kilometre, there are a total of two ant nests and/or beehives (i.e., S = 2). (d) [5 marks] Suppose that a myrmecologist walks through the region searching for ant nests, and a melittologist also walks through the region searching for beehives. Let / be the area covered until the myrmecologist finds the second ant nest, and H be the area covered until the melittologist finds the first beehive. Identify and justify, which distributions N and H follow. (e) [5 marks) Derive the expressions for probability distribution functions of N and H from the probabilities of observing certain counts of ant nests and beehives. Hint: Counts of nests/beehives and the area covered to find nests/beehives are different variables. Note: you may not use the answer from (d) to simply state what the distribution is. You must derive the same answer independently from (d). (f) [5 marks] Show that the joint probability density function of N and H is given by fNH (n, h) = 0.108 ne-0.6n-0.3h, n >0,h > 0 (g) [5 marks) Calculate the value of P(H > 2N). Hint: You will need to use integration by parts to obtain this probability. (h) [10 marks] Use Python to empirically estimate the value of P(H > 2N) . To do this: Randomly generate 100,000 values from N using the appropriate function Randomly generate 100,000 values from H using the appropriate function Calculate a vector of differences H - 2/ and find the proportion of these differences which are greater than 0. Provide both the Python statements and their output