Suppose you are interested in determining the proportion, p, of current Bond students who agree with the statement: \"Math Stats is the most best-est class ever\". To investigate, you stand in the middle of the quadrangle one afternoon and ask passing students. Let X ,- represent the it\" student's response, with X,- = 1 if they agree with the statement and X,- = 0 otherwise. So, X,- iid Bern(p). Further, let N ~ P0is(100) be the number of responses you get over the afternoon, so the total number of positive responses is S = 21iv=1 X i. Finally, assume N is independent of the X ,- 's, making 5 a random sum. (a) Find the mean and variance of S in terms of p. [2 marks] (b) Show that, in this case, 5 ~ Pois(100p). [HINT: Think generating functions!] [1 mark] One obvious estimator of p would be S / N . However, this estimate has the issue that it is undened in the case that N = 0. To deal with this issue, we propose two different estimators: T1 = 5/100 and T2 = S/(N + 1) [NOTE: You may nd the following facts about N useful: 2 N N N ElNH} 0.9900, E{W} 0.0099, E {W} 0.9801 and you may use them to answer the following questions] (c) Are either of these estimators unbiased? Explain your answers. [2 marks] (d) Which of these two estimators would you prefer? Why? [1 mark]