Probability | Part IA, 2003

The random variables XX and YY each take values in {0,1}\{0,1\}, and their joint distribution p(x,y)=P{X=x,Y=y}p(x, y)=P\{X=x, Y=y\} is given by

p(0,0)=a,p(0,1)=b,p(1,0)=c,p(1,1)=d.p(0,0)=a, \quad p(0,1)=b, \quad p(1,0)=c, \quad p(1,1)=d .

Find necessary and sufficient conditions for XX and YY to be (i) uncorrelated; (ii) independent.

Are the conditions established in (i) and (ii) equivalent?

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