Assume three binary random variables

a, b, c 2 f0, 1g have the following joint distribution:

By direct evaluation, show that this distribution has the property that a and c are marginally dependent (i.e., p¹a, cº , p¹aºp¹cº), but a and c become independent when conditioned on b (i.e., p¹a, cjbº = p¹ajbºp¹cjbº). Based on this joint distribution, draw all possible directed graphs for

a, b,

c, and compute all conditional probabilities for each graph.

a b c | p(a,b,c) 0 00 0.024 0 0 1 0.056 0 1 0 0.108 0 1 1 0.012 1 0 0 0.120 1 0 1 0.280 1 1 0 0.360 1 1 1 0.040