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 independent (i.e., p¹a, cº = p¹aºp¹cº), but a and c become dependent 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.

Transcribed Image Text:

a bc cp(a,b,c) 0 00 0.072 0 0 1 0.024 0 1 0 0.008 0 1 1 0.096 1 00 0.096 1 0 1 0.048 1 1 0 0.224 1 1 1 0.432