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 are independent of other beehives, and the locations of ant nests and beehives are independent of each other.

a) Let A 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? = 0.6 and B follows a Poisson distribution with ? = 0.3.

b) 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 S follows a Poisson distribution with ? = 0.9.

Show that the joint probability density function of N and H is given by fNH (n, h) = 0.108 ne-O.6n-0.3h n > 0, h >0Note: A Poisson random variable X with parameter 1 has moment generating function: My (t) = exp [1(et - 1)]