# Why the covariance is zero for independent variables?

when the two independent variables are uncorrelated the covariance seems to be zero. Is there any equation for it which describes the fact?

• en.wikipedia.org/wiki/Covariance_and_correlation Jun 10, 2018 at 8:23
• i couldn't found any solving equations about zero covariance.Can you please help? Jun 10, 2018 at 8:43
• Just expand the product in the definition of covariance and use independence to see that covariance is 0 Jun 10, 2018 at 8:45
• Ok, I posted an answer with the details Jun 10, 2018 at 8:48

## 2 Answers

If you know that $\rho_{X Y} = \sigma_{X Y} / (\sigma_X \sigma_Y)$ is zero, then $\sigma_{X Y}$ must be zero.

Note that the covariance of two independent variables is $\sigma_{X Y} = E[(X-EX)(Y-EY)] = E[X Y] - E[X] E[Y] = 0$, because by independence $E[X Y] = E[X] E[Y]$.

yes, definitely if the two random variable is independent then the covariance is zero. suppose X and Y be two independent random variable then occurrence of X or Y does affect the occurrence of Y

i.e

P(X/Y) = P(X) and P(Y/X) = P(Y)

i.e

P(XY) = P(X)P(Y)

also we know that

COV(XY) = E[X-E(X)] [Y-E(Y)]

    = E[XY] - E[X] E[Y]
= ΣXYP(XY) - ΣXP(X) ΣYP(Y)
= ΣXYP(X)P(Y) - ΣXYP(X)P(Y)
= 0


Here ΣXY = ΣX ΣY

(as these are the finite convergent series.)