Show that this is an example for which X and Y have zero cor

Show that this is an example for which X and Y have zero correlation but are not independent.

Solution

E(x) = 2p(-1)+0+1(2p) =0

E(y) =0

As E(x) and E(Y) =0

E(XY) = p(-1)+0+p(1)+(1-4p)0+p(-1)+p =0

Hence cov (xy) = E(XY)-E(x)E(y) =0

Hence correlation r =0 as numerator =0

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But x and y are not independent.

Consider P(x=-1, y =1) = p

P(X=-1) = 20 and P(Y=1) =2p

Hence P(XY) not equals P(X)P(Y)

Therefore x and y are not independent though r =0

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a) E(Xy^2) = 0.10(0)+0.15(1)+0.25(0) +0,50(4)

= 2.15

b)E(max xy) = 4(0)+4(0+4(0.5) = 2