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[See] Show that if X and Y are 2 -state random variables and Cov;(X, Y)=0 then X and Y are independent. That is, if a, b, c, d are real numbers with P(X=a)=p_X,

Question: Show that if \(\mathrm{X}\) and \(\mathrm{Y}\) are 2 -state random variables and \(\operatorname{Cov}(X, Y)=0\) then \(\mathrm{X}\) and \(\mathrm{Y}\) are independent. That is, if \(\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\) are real numbers with \(P(X=a)=p_{X}, P(X=b)=1-p_{X}\) and

\(P(X=c)=p_{Y}, P(X=d)=1-p_{Y}\)

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