Now,

g(x)=∑

_{y}y×f(x,y) for given value of x

h(y)=∑

_{x}x×f(x,y) for given value of y

Now,

Cov(X,Y) = E(XY) – E(X) E(Y)

Now,

E(X,Y)=∑

_{x}∑

_{y }xy f(x,y)

E(X,Y)=∑

_{x}∑

_{y}xy g(x).h(y)

E(X,Y)=∑

_{x }x g(x) ∑

_{y}y h(y)

E(X,Y)=E(X) E(Y)

So,

Cov(X,Y) = E(XY) – E(X) E(Y)

Cov(X,Y) = E(X) E(Y) – E(X)E(Y)

Cov(X,Y) =0.

Hence,

It is proved that the covariance of two independent random variable is 0.

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