a) Let $X$ and $Y$ be two uncorrelated random variables. Assume $Var(X) = 1.55$ and $Var(Y) = 0.8$. What is the variance of the random variable $Z = -4X + 5Y - 6$?

b) What if $X$ and $Y$ are correlated with $Cov(X,Y) = 0.6$?

For a) since the variables are uncorrelated I thought you could just sum the variances of the variables? So $$Var(Z) = -4Var(X) + 5Var(Y)$$ However this produces an incorrect answer.

  • 2
    $\begingroup$ So you think the variance of $-4X$ is $-4$ times the variance of $X$? Nothing preoccupying there? $\endgroup$ – Did Feb 6 '19 at 10:49
  • $\begingroup$ I don't see what else it would be? Im new to statistics so Im probably missing something obvious. $\endgroup$ – Pame Feb 6 '19 at 11:34
  • $\begingroup$ "Im probably missing something obvious" Indeed you are! Recall that every variance is nonnegative. $\endgroup$ – Did Feb 6 '19 at 18:42

Watch out! While it is true that the variance of a sum of uncorrelated random variables is indeed the sum of their variances, as Did suggested you must give special attention to the constants scaling the random variables.

In general, for a random variable $X$ and for $a,b\in\mathbb{R}$, we have that

$$Var(a+bX) = E[(a+bX) - (a+bE(X))^2] = E[b^2(X-E(X))^2] = b^2Var(X)$$

which means that variance is scaled quadratically wrt its random variable's multiplicative constant, and not at all with respect to additive constants. Can you figure it out from here?

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