4
$\begingroup$

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

$\endgroup$
4
  • $\begingroup$ en.wikipedia.org/wiki/Covariance_and_correlation $\endgroup$
    – Miguel
    Jun 10, 2018 at 8:23
  • $\begingroup$ i couldn't found any solving equations about zero covariance.Can you please help? $\endgroup$ Jun 10, 2018 at 8:43
  • 1
    $\begingroup$ Just expand the product in the definition of covariance and use independence to see that covariance is 0 $\endgroup$
    – Miguel
    Jun 10, 2018 at 8:45
  • $\begingroup$ Ok, I posted an answer with the details $\endgroup$
    – Miguel
    Jun 10, 2018 at 8:48

2 Answers 2

7
$\begingroup$

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 $E[X Y] = E[X] E[Y]$.

$\endgroup$
0
$\begingroup$

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.)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.