Researchers at the University of Kent (England) developed models for counting the number of colonies of microorganisms in liquid (Journal of Agricultural, Biological, and Environmental Statistics, June 2005). The expected number of colonies was derived under the “inhibition” model. Consider a microorganism of species A. Suppose n species A spores are deposited in a Petri dish, and let Y equal the number of species A spores that grow. Then Y = ∑_i=1ⁿ x_i where x_i = 1if a species A spore grows and x_i = 0 if a species A spore is inhibited from growing.
a. Show that E (Y) = np, where p = P(x_i = 1). .
b. In an alternative model, the researchers showed that p = P(x_i = 1) depends on the amount S of soil in the Petri dish, where p = e^-θs and θ is a constant. In all their experiments, the amount of soil placed in the Petri dish was essentially the same. For these experiments, explain why E(Y) = ne^-θs.