If (X,Y ) has a bivariate normal distribution, find the conditional
distribution of Y given X = x
Consider a doctor pondering the following dilemma: “If I’m at least 80
percent certain that my patient has this disease, I will recommend survey;
whereas if I’m not quite as certain, then I recommend additional tests that
are expensive and sometimes painful.” Now, initially I was only 60 percent
certain that Mr. Jones has the disease, so I ordered the Test A, which
always gives a positive result when the patient has the disease and almost
never gives a positive result when he is healthy. The test result was
positive, and I was all set to recommend surgery. However, Mr. Jones
informed me, for the first time, that he is a diabetic (糖尿病患者). This
information complicates this decision because, although it doesn’t change
my original 60 percent estimate of his changes of having the disease, it
does affect the interpretation of the results of the Test A since Test A,
while never yielding a positive result when the patient is healthy, does
unfortunately yield a positive result 30 percent of the time in the case of
diabetic patients not suffering from the disease. Now what should the
doctor do? More tests or immediate surgery? (20%)
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