1. The "capture-recapture" method is often used to estimate the size, N, of a wildlife population. This problem is about the related question of estimating a human population (homeless persons) when it is not ethical to use the usual "capture-recapture" methodology of tagging. This problem comes from a real life study design implemented in the Toronto area for this purpose, which was discussed in class. Suppose that 20 actors are hired to impersonate homeless persons and asked to mix as much as possible with the homeless population of Toronto. Next, surveyors are sent into the eld to identify a random sample of 50 seemingly homeless persons. If, during questioning, an seemingly homeless person states that they are a "decoy" (actor), the surveyor must note this fact. Let Y represent the number of apparently homeless persons surveyed that are found to be decoys and let M be the total number of homeless persons (not including actors) in the population. Please note: the solution is not a direct copy of the "capture-recapture" example we did in class, though it is related. You must think carefully about the problem to obtain the correct probability model. [Total 20 pts]

(a) Write the expression for the probability of observing y = 5 decoys in the sample. (8 pts)

(b) If the size of the homeless person population (not including decoys) is M = 200, how many decoys do we expect to observe in our sample? How variable is the number decoys surveyed? Hint: Y has a named distribution. (6 pts)

(c) A sociologist guesses that the total number of homeless in the city (excluding decoys) is either M = 150, M = 200 or M = 250. What are the probabilities of surveying y = 5 decoys under each scenario? Briefly state which of these estimates of the total population you would choose and why? (6 pts)