Editing Covariance (section)

==Examples==
Consider three independent random variables <math>A, B, C</math> and two constants <math>q, r</math>.
<math display="block">
\begin{align}
X &= qA + B \\
Y &= rA + C \\
\operatorname{cov}(X, Y)
&= qr \operatorname{var}(A)
\end{align}
</math>
In the special case, <math>q=1</math> and <math>r=1</math>, the covariance between <math>X</math> and <math>Y</math> is just the variance of <math>A</math> and the name covariance is entirely appropriate.

[[File:Covariance_geometric_visualisation.svg|thumb|300px|Geometric interpretation of the covariance example. {{nowrap|Each cuboid is the}} [[axis-aligned]] [[bounding box]] of its point {{nowrap|(''x'', ''y'', ''f''&thinsp;(''x'', ''y'')),}} and the {{nowrap|''X'' and ''Y'' means}} (magenta point). {{nowrap|The covariance}} is the sum of the volumes of the cuboids in the 1st and 3rd quadrants (red) and in the 2nd and 4th (blue).]]
Suppose that <math>X</math> and <math>Y</math> have the following [[joint probability distribution|joint probability mass function]],<ref>{{Cite web| url=https://onlinecourses.science.psu.edu/stat414/node/109| title=Covariance of X and Y {{!}} STAT 414/415|publisher=The Pennsylvania State University | access-date=August 4, 2019 | archive-url=https://web.archive.org/web/20170817034656/https://onlinecourses.science.psu.edu/stat414/node/109 | archive-date=August 17, 2017}}</ref> in which the six central cells give the discrete joint probabilities <math>f(x, y)</math> of the six hypothetical realizations {{nowrap|<math>(x, y) \in S = \left\{ (5, 8), (6, 8), (7, 8), (5, 9), (6, 9), (7, 9) \right\}</math>:}}
{| class="wikitable" style="text-align:center;"
!rowspan="2" colspan="2"|<math>f(x,y)</math>
!colspan="3"|''x''
|rowspan="6" style="padding:1px;"|
!rowspan="2"|<math>f_Y(y)</math>
|-
!5
!6
!7
|-
!rowspan="2"|''y''
!8
|0
|0.4
|0.1
|0.5
|-
!9
|0.3
|0
|0.2
|0.5
|-
|colspan="7" style="padding:1px;"|
|-
!colspan="2"|<math>f_X(x)</math>
|0.3
|0.4
|0.3
|1
|}
<math>X</math> can take on three values (5, 6 and 7) while <math>Y</math> can take on two (8 and 9). Their means are <math>\mu_X = 5(0.3) + 6(0.4) + 7(0.1 + 0.2) = 6</math> and <math>\mu_Y = 8(0.4 + 0.1) + 9(0.3 + 0.2) = 8.5</math>. Then,
<math display="block">\begin{align}
  \operatorname{cov}(X, Y)
    ={} &\sigma_{XY} = \sum_{(x,y)\in S}f(x, y) \left(x - \mu_X\right)\left(y - \mu_Y\right) \\[4pt]
    ={} &(0)(5 - 6)(8 - 8.5) + (0.4)(6 - 6)(8 - 8.5) + (0.1)(7 - 6)(8 - 8.5) +{} \\[4pt]
        &(0.3)(5 - 6)(9 - 8.5) + (0)(6 - 6)(9 - 8.5) + (0.2)(7 - 6)(9 - 8.5) \\[4pt]
    ={} &{-0.1} \; .
\end{align}</math>