Question 1: 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 student's response, vith X = 1 if they agree with the statement and X = 0 otherwise. So, X iid Bern(p). Further, let V- Pois(100) be the number of responses you get over the afternoon, so the total number of positive responses is S = E-1X. Finally, assume N is independent of the X,'s, making S a random sum. (a) Find the mean and variance of S in terms of p. (b) Show that, in this case, S Pois(100p). [HINT: Think generating functions!] One obvious estimator of p would be S/N. However, this estimate has the issue that it is undefined in he case that N = 0. To deal with this issue, we propose two different estimators: %3D %3D %3D C (100
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![Question 1:
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 i student's response,
with X = 1 if they agree with the statement and X = 0 otherwise. So, X, iid Bern(p). Further, let
N - Pois(100) be the number of responses you get over the afternoon, so the total number of positive
responses is S = E-1 X1. Finally, assume N is independent of the X,'s, making S a random sum.
(a) Find the mean and variance of S in terms of p.
(b) Show that, in this case, S Pois(100p). [HINT: Think generating functions!]
One obvious estimator of p would be S/N. However, this estimate has the issue that it is undefined in
the case that N = 0. To deal with this issue, we propose two different estimators:
%3!
%3D
T1 = S/100 and T2 = S/(N + 1)
NOTE: You may find the following facts about N useful:
N
N
N2
3= 0.900; ED=
= 0.0099; E-
= 0.9801
E
IN+1)
%3D
(N+1)2
(N + 1)2
and you may use them to answer the following questions.]
(c) Are either of these estimators unbiased? Explain your answers.
(d) Which of these two estimators would you prefer? Why?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F50808ef8-5f35-464a-999f-8aa25e4c15a1%2F03984998-1263-437c-a757-9e259553803d%2Fhz1ult_processed.jpeg&w=3840&q=75)
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