Editing Variance (section)

=== Linear combinations ===
In general, for the sum of <math>N</math> random variables <math>\{X_1,\dots,X_N\}</math>, the variance becomes:
<math display="block">\operatorname{Var}\left(\sum_{i=1}^N X_i\right) = \sum_{i,j=1}^N\operatorname{Cov}(X_i,X_j) = \sum_{i=1}^N\operatorname{Var}(X_i) + \sum_{i,j=1,i\ne j}^N\operatorname{Cov}(X_i,X_j),</math> 
see also general [[Bienaymé's identity]].

These results lead to the variance of a [[linear combination]] as:

<math display="block">\begin{align}
\operatorname{Var}\left( \sum_{i=1}^N a_iX_i\right) &=\sum_{i,j=1}^{N} a_ia_j\operatorname{Cov}(X_i,X_j) \\
&= \sum_{i=1}^N a_i^2 \operatorname{Var}(X_i) +   \sum_{i \neq j}            a_i a_j \operatorname{Cov}(X_i,X_j)\\
&= \sum_{i=1}^N a_i^2 \operatorname{Var}(X_i) + 2 \sum_{1 \leq i < j \leq N} a_i a_j \operatorname{Cov}(X_i,X_j).
\end{align}</math>

If the random variables <math>X_1,\dots,X_N</math> are such that
<math display="block">\operatorname{Cov}(X_i,X_j)=0\ ,\ \forall\ (i\ne j) ,</math>
then they are said to be [[Covariance#Definition|uncorrelated]]. It follows immediately from the expression given earlier that if the random variables <math>X_1,\dots,X_N</math> are uncorrelated, then the variance of their sum is equal to the sum of their variances, or, expressed symbolically:

<math display="block">\operatorname{Var}\left(\sum_{i=1}^N X_i\right) = \sum_{i=1}^N\operatorname{Var}(X_i).</math>

Since independent random variables are always uncorrelated (see {{Section link|Covariance|Uncorrelatedness and independence}}), the equation above holds in particular when the random variables <math>X_1,\dots,X_n</math> are independent.  Thus, independence is sufficient but not necessary for the variance of the sum to equal the sum of the variances.

====Matrix notation for the variance of a linear combination====

Define <math>X</math> as a column vector of <math>n</math> random variables <math>X_1, \ldots,X_n</math>, and <math>c</math> as a column vector of <math>n</math> scalars <math>c_1, \ldots,c_n</math>. Therefore, <math>c^\mathsf{T} X</math> is a [[linear combination]] of these random variables, where <math>c^\mathsf{T}</math> denotes the [[transpose]] of <math>c</math>. Also let <math>\Sigma</math> be the [[covariance matrix]] of <math>X</math>. The variance of <math>c^\mathsf{T}X</math> is then given by:<ref>{{Cite book | last1=Johnson | first1=Richard | last2=Wichern | first2=Dean | year=2001 | title=Applied Multivariate Statistical Analysis | url=https://archive.org/details/appliedmultivari00john_130 | url-access=limited | publisher=Prentice Hall | page=[https://archive.org/details/appliedmultivari00john_130/page/n96 76] | isbn=0-13-187715-1 }}</ref>

<math display="block">\operatorname{Var}\left(c^\mathsf{T} X\right) = c^\mathsf{T} \Sigma c .</math>

This implies that the variance of the mean can be written as (with a column vector of ones)

<math display="block">\operatorname{Var}\left(\bar{x}\right) = \operatorname{Var}\left(\frac{1}{n} 1'X\right) = \frac{1}{n^2} 1'\Sigma 1.</math>