Question

Suppose 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 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 λ = 0.9.

(b) Find the probability that in a randomly chose square kilometre, there are a combined 2 ant nests and beehives (i.e. S = 2).

(c) Suppose that an ecologist walks through the region searching for ant nests, and a second ecologist also walks through the region searching for beehives. Let N be the distance that the myrmecologist walks until finding the first ant nest, and H be the distance the melittologist walks until finding the first beehive. Justify why N follows an exponential distribution with λ = 0.6 and H follows an exponential distribution with λ = 0.3.

Answer 1