Let X and Y are two independent random variables, then f(x,y)=g(x).h(y)—-1

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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