Suppose P(X = 1, Y = 1) = 0.6, P(X = 2, Y = 1) = 0.2, and P(X = 1, Y = 2) = 0.2. Find Cov(X, Y ) and the correlation ρ of X and Y .?

P( X = 1 ) = P( X = 1 , Y = 1 ) + P( X = 1 , Y = 2 ) = 0.6 + 0.2 = 0.8
P( X = 2 ) = P( X = 2 , Y = 1 ) = 0.2
μ(X) = E(X) = Σ xP(X) = 1*0.8 + 2*0.2 = 1.2
E(X²) = Σ x²P(X) = 1²(0.8) + 2²(0.2) = 1.6
V(X) = E(X²) - μ² = 1.6 - 1.2² = 0.16
σ(X) = √ V(X) = √ 0.16 = 0.4

P( Y = 1 ) = P( X = 1 , Y = 1 ) + P( X = 2 , Y = 1 ) = 0.6 + 0.2 = 0.8
P( Y = 2 ) = P( X = 1 , Y = 2 ) = 0.2
μ(Y) = E(Y) = Σ yP(Y) = 1*0.8 + 2*0.2 = 1.2
E(Y²) = Σ y²P(Y) = 1²(0.8) + 2²(0.2) = 1.6
V(Y) = E(Y²) - μ² = 1.6 - 1.2² = 0.16
σ(Y) = √ V(Y) = √ 0.16 = 0.4

E(XY) = Σ xyP(X,Y) = 1*1*0.6 + 2*1*0.2 + 1*2*0.2 = 1.4
Cov ( X , Y ) = E(XY) - E(X)E(Y) = 1.4 - 1.2*1.2 = - 0.04
ρ = Cov ( X , Y ) / [ σ(X) * σ(Y) ] = - 0.04 / [ 0.4 * 0.4 ] = - 0.25

Ans :
Cov ( X , Y ) = - 0.04
ρ = - 0.25