Exercise 1 (4 points): Consider the example of Chapter

Exercise 1 (4 points):
Consider the example of Chapter II in page 90-91.
Suppose we observe X = i females in the sample of size n (n = 5). What is the posterior
probability P(D = jjX = i) in the whole population of size M = 20? Give the mathematical
formula.
Exercise 2 (4 points):
Do Exercise 0.9 in Chapter II in page 95.
Hint: Write
P(X_ = x) = P(fX_ = xg f[j(_ = j)g)
and use rules of probability calculus.
Exercise 3 (4 points):
Do Exercise 0.13 in Chapter IV in page 144.
Exercise 4 (4 points):
Consider the table of conditonal probability of Dyspnea given Tuberculosis or Lung Cancer and
Bronchitis in Chapter IV page 148. We call
T = event Tuberculosis is present
L = event Lung Cancer is present
B = event Bronchitis is present
Assume P(T [ L) =0.0648 and P(B) = 0.45.
Assume that the events T [ L and B are independent. Compute the probability of Dyspnea to
be present. This event is denoted by D.
Exercise 5 (4 points):
Consider the Bayesian network of Chapter IV Figure 14 page 149. We do not have T [L and B
independent since L and B have a common parent which is smoking. We denote by S = Smoking.
We assume
T is independent from L or from B
L and B given S are independent
We assume the Table 6 in page 148 holds and
P(T) = 0:01 P(S) = 0:5 P(LjS) = 0:1 P(LjS) = 0:01; P(BjS) = 0:6 P(BjS) = 0:3:
Compute again P(D) = P(Dyspnea):
Hint: Compute .rst P(L) P(B) and P(T [ L B).
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