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This assignment will strengthen your understanding of the fundamental concepts in applied
probability and statistics. The questions in this assignment will build on lecture material as
well as the assigned readings from (Wasserman, L. 2004. \All of Statistics”).

Deliverables Submit your responses as a PDF along with code to D2L by the stated
deadline. Show all work along with answers. This is for your bene t as incorrect
answers may receive partial credit if the work demonstrates understanding.

Problem 1: Conditional Independence (2 points)

Let A;B;C 2 f0; 1g be three binary random variables with the following joint probability

A B C P(A;B;C)
0 0 0 0.192
0 0 1 0.144
0 1 0 0.048
0 1 1 0.216
1 0 0 0.192
1 0 1 0.064
1 1 0 0.048
1 1 1 0.096

a) By direct calculation, compute the marginal P(A;B).

b) By direct calculation compute the marginals P(A) and P(B).

c) Are the random variables A and B dependent? Why or why not?

d) Compute the conditional P(A;B j C)

e) Show that A and B are conditionally independent, given C.

Problem: Diagnostic Tests and Baye’s Rule (1 point)

I have decided to get myself tested for COVID-19 antibodies. However, being comfortable
with statistics, I am curious about what the test means for my actual status. Let’s investigate
these questions, showing all your work:

a) According to the FDA, the UA COVID-19 antibody test (known as ELISA pan-Ig) has a
sensitivity (a.k.a. true positive rate) of 97.5% and a speci city (a.k.a. true negative rate)
of 99.1%. Assume that 5% of the population actually have COVID-19 antibodies. Write
down the joint probability (prior/likelihood) with events for disease state S 2 ftrue; falseg
and test result R 2 ftrue; falseg.

b) Assuming I receive a positive test result, use Bayes’ rule to calculate the probability that
I actually have COVID-19 antibodies?

c) Assuming I receive a negative test result, what is the probability that I do not have
COVID-19 antibodies?

d) Assume I take the test twice, and receive a positive result in both tests. What is the
probability that I have COVID-19 antibodies according to Bayes’ rule?

e) Now assume that only 1% of the population has COVID-19 antibodies. Repeat parts (b)
and (c) with this revised prior belief.

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