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)]