Prove that the covariance of two independent random variable is 0.

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.