# Variance of Uncorrelated Variables

I know the typical variance formula for correlated random variables, but can't seem to find the variance for a linear combination of uncorrelated random variables.

Now there are a few things regarding uncorrelated variables that obviously play into this: - Two random variables are said to be uncorrelated if their Cov(X,Y)=0 - The variance of the sum of uncorrelated random variables is the sum of their variances

But what about the variance itself for a linear combination of these r.v.'s?

• What is the question? You seem to know everything already Sep 19, 2016 at 20:36
• Is there a formula for the variance of a linear combination of uncorrelated random variables? I know the rules regarding variances, but can't seem to find the formula for a linear combination of these r.v.'s anywhere. Sorry, I edited my question to include "linear combination" Sep 19, 2016 at 20:43
• It is the sum of the individual variances
– Carl
Sep 19, 2016 at 20:43
• for two variables, see stats.stackexchange.com/a/123963/805 or more generally en.wikipedia.org/wiki/Variance#Basic_properties (then taking advantage of the fact that when the covariances are 0 the terms including them will drop out) Sep 20, 2016 at 4:25
• Possible duplicate of Use the properties of linear combinations to derive means and standard deviations
– Tim
Feb 27, 2017 at 19:11

Let $Z$ be a linear combination of two random uncorrelated variables $X$ and $Y$ so that:

$$Z=aX+bY$$

Then:

$$var(Z)=a^2·var(X)+b^2·var(Y)$$

There's some confusion. "Variance" is not a property of a pair of variables, it's a property of a random variable. If your r.v. happens to be the sum of two others, then there is a formula for that variance as a function of the other two. It depends on the correlation, and if that correlation is zero, then plug in zero, and there you go. To take from Pere's answer, if $$Z = aX+bY$$ then $$\newcommand{\var}{{\rm var}} \var(Z)=a^2·\var(X)+b^2·\var(Y) + 2·{\rm cov}(X, Y)$$

• Isn't a factor $a.b$ needed before the covariance term? Feb 25, 2017 at 17:48
• That was my own answer. "Filipe" was just the profile I was using on another machine. (for boring reasons).
– f.g.
Feb 28, 2017 at 20:13

If $X_1, ..., X_K$ are all uncorrelated with each other, then

$${\rm var} \left( \sum_{i=1}^K a_i X_i \right) = \sum_{i=1}^{K} a_i^2 {\rm var}(X_i)$$

• The previous two answers pre-specified the case where $K=2$ Feb 27, 2017 at 19:04