1. In a population of women, a woman is pregnant with probability A. A new test for pregnancy has been developed. The probability that a pregnant woman tests positive is 9 and probability that a non-pregnant woman tests positive is 17. (a) (C) (d) (8) A scientist conducts a small trial to estimate the value of 9 by giving the new pregnancy test to 10 pregnant women. There were 9 positive tests. The scientist wishes to use a Bayesian analysis and suggests a Beta(a, 5) prior. Derive the posterior distribution of 6 and show that it can be written as a Beta distribution. [5] The scientist wants to choose values for a and . She asks two colleagues who have an indepth knowledge of pregnancy tests to provide some information. One suggests that from 20 tests used by pregnant women, he believes that 17 would be positive. The second suggests that from 30 tests used by pregnant women, she believes that 29 would be positive. How should the scientist use this information to choose a and B. [4] Now the scientist wishes to predict the number (denoted by g) of positive results out of a new group of 10 pregnant women who are going to take the test. Derive the predictive distribution p(g'}|y). [6] A woman decides to use the test to see whether she is pregnant. She tests negative. What is the probability that she is in fact pregnant? [4] The same woman, having seen that the rst test was negative, decides to repeat the test, in order to provide more reassurance that she is not pregnant. Assuming that repeat tests on the same woman are conditionally independent given her true pregnancy status, what is the probability that the second test will also be negative? [6] You may use the following notation and results: The Binomial distribution with success probability 6, Binomial(n,t9), has probability mass function p(yl6)=(:)9\"(16)"y, y=0,1,...,n; 05651. The Beta distribution, Beta(a, ,8), has probability density function NO\"- + 3) 1 ,81 p (1,6 0' 1 , 0 22m. p